University  of  California  •  Berkeley 

The  Theodore  P.  Hill  Collection 

of 
Early  American  Mathematics  Books 


Digitized  by  the  Internet  Archive 

in  2008  with  funding  from 

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http://www.archive.org/details/easyalgebraforbeOOvenarich 


AN 


EASY    ALGEBRA 


FOE    BEGINNERS; 


BEING  A  SIMPLE,   PLAIN  PRESENTATION   OF   THE 

ESSENTIALS  OF  ELEMENTARY  ALGEBRA,  AND 

ALSO  ADAPTED  TO  THE  USE  OF  THOSE 

WHO  CAN   TAKE  ONLY  A  BRIEF 

COURSE     IN     THIS     STUDY. 


BY 


CHARLES  S.  VENABLE,  LL.D., 

Professor  op  Mathematics  in  the  University  op  Virginia,  and  Author 
op  A  Sekies  of  Mathematical,  Text-Books. 


NEW  YORK: 

UNIVERSITY   PUBLISHING   COMPANY. 

1886 


Copyright, 

1880, 

Bt  university  publishing  company. 


b*878 


PREFACE 


This  book  is  designed  for  the  use  of  those  for  whom  the 
High  School  (Elementary)  Algebra  may  be  too  difficult, 
and  is  adapted  also  to  those  who  can  take  only  a  brief 
course.  It  has  been  carefully  prepared  with  a  view  to 
render  an  acquaintance  with  the  essentials  of  elementary 
algebra  easy  of  acquisition  by  the  young  beginner.  The 
explanations  are  brief  and  simple.  The  examples  are  not 
difficult. 

Such  definitions  and  rules  as  are  common  to  arithmetic 
and  algebra  have  been  given  without  labored  illustration, 
or  have  been  assumed  as  already  familiar  to  the  pupil. 

As  a  preparation  for  the  solution  of  problems  by  means 
of  equations,  well  graded  steps  are  given  in  the  section 
preceding  these  problems  for  practice  in  the  translation 
of  quantitative  statements  from  ordinary  language  into 
algebraic  expressions. 

Only  the  leading  and  more  easily  understood  principles 
of  Radicals  and  Progressions  have  been  introduced,  and  I 
trust  that  these  more  advanced  topics  are  presented  in  a 
simple  and  attractive  form. 


iy 


PREFACE. 


The  Miscellaneous  Examples  for  independent  exercise 

on   the   subjects   of   the   different  sections  will  give  the 

pupil   greater  familiarity  with  the  methods  learned  in  the 

text ;  while  the  Review  Questions  will  serve  as  a  test  of 

the  accuracy  of  his  knowledge  of  the  principles  underlying 

these  methods  and  operations. 

0.  S.  V. 

University  of  Virginia, 
November  10,  1880. 


CONTENTS. 


PAGE 

I.  Definitions 7 

II.  Addition 10 

III.  Subtraction 12 

IV.  Multiplication 14 

V.  Brackets  or  Parentheses 18 

VI..  Multiplication  by  Inspection 20 

VII.  Division 23 

VIII.  Factoring 28 

IX.  Greatest  Common  Divisor 30 

X.  Least  Common  Multiple 32 

XI.  Reduction  of  Fractions  to  Lowest  Terms 33 

XII.  Reduction  of  Fractions  to  a  Common  Denominator.  35 

XIII.  Addition  of  Fractions 37 

XIV.  Subtraction  of  Fractions , 39 

XV.  Multiplication  of  Fractions , 41 

XVI.  Division  of  Fractions 43 

XVII.  Finding  Numerical  Values  by  Substitution 44 

XVIII.  Simple  Equations 46 

XIX.  Translation  of  Ordinary  Language  into  Algebraic 

Expressions 53 

v 


vi  CONTENTS. 

PAGE 

XX.  Problems  in  Simple  Equations 56 

XXI.  Problems — Continued GO 

XXII.  Simple  Equations  with  Two  Unknown  Quantities.— 

Elimination G3 

XXIII.  Problems  producing  Simultaneous  Equations 67 

XXIV.  Simultaneous  Simple  Equations  of  Three  or  More 

Unknown  Quantities 70 

XXV.  Involution  or  Raising  to  Powers 73 

XXVI.  Evolution  or  Extraction  of  Roots. — Square  Hoot.  76 

XXVII.  Quadratic   Equations 84 

XXVIII.  Solution  of  Affected  Equations 86 

XXIX.  Problems  giving  rise  to  Quadratic  Equations 89 

XXX.  Easy  Simultaneous  Equations  solved  by  Quadrat- 
ics      91 

XXXI.  Radicals  of  the  Second  Degree 94 

XXXII.  Ratio   and  Proportion 101 

XXXIII.  Arithmetical  Progressions 106 

XXXIV.  Geometrical  Progressions Ill 

XXXV.  Miscellaneous  Examples 117 

XXXVI.  General  Review  Questions 134 

Answers 143 


EASY    ALGEBRA. 


SECTION    I. 

DEFINITIONS. 

1.  Algebra. — In  algebra  we  use  letters  to  denote  num- 
bers, and  signs  to  indicate  the  arithmetical  operations  to 
be  performed  on  them. 

2.  Signs. — The  usual  signs,  some  of  which  we  have 
used  in  arithmetic,  are  : 

(1.)  The  sign  of  addition  +  (plus),  as  a  -f  b. 

(2.)  The  sign  of  subtraction  —  (minus),  as  a  —  b. 

(3.)  The  signs  of  multiplication  x,  •  ,  and  simply  writ- 
ing letters  one  after  another,  as  a  x  b}  a  •  b,  and  ab, 
all  mean  a  multiplied  by  b. 

So,  also,  5  x  a,  5  •  a,  and  5a  mean  5  times  a. 

Note.— In  writing  figures  we  must  remember  that  57  means  5  and  yL,  and 
57  means  50  +  7,  and  not  5  x  7. 

(4.)  The  signs  of  division  -r-,  and  a  line  between  the 
letters.     Thus, 

a  -7-  b,  and  -r   mean  a  divided  by  b. 
(5.)  The  sign  of  equality  =,  read  "is  equal  to."   Thus, 
-r  =  c  is  read  "  a  divided  by  b  is  equal  to  c." 


8  DEFINITIONS. 

3.  Coefficient.— As  6  +  6  +  6  +  6  +  6  is  5x6.  So 
#  +  #  +  #  +  «  +  «  is  5  x  a,  which  we  write  5a,  and  5  is 
called  the  coefficient  of  a.  Similarly,  ab  +  ab  +  ab  +  ab  + 
ab  -h  ab  =  Gab,  and  6  is  the  coefficient  of  the  product  ab. 

Definition. — The  coefficient  is  the  number  ivritten  be- 
fore a  letter  or  quantity  to  show  the  number  of  times  it  is 
taken.  When  no  coefficient  is  written,  the  coefficient  1  is 
understood. 

4.  Algebraic  Quantity,  or  Algebraic  Expres- 
sion.— Any  collection  of  letters  with  algebraic  signs  is 
called  an  algebraic  quantity,  or  an  algebraic  expression. 
Thus, 

a,  a  +  b  +  c  —  d,  ab,  hab  +  2cd  —  3ef 

are  algebraic  expressions. 

5.  Terms. — The  terms  of  an  algebraic  expression  are 
the  different  parts  separated  by  the  sign  +  or  — .  Thus, 
in  the  expressions  a  +  b  +  c  —  d,  hab  +  2cd  —  3ef,  a,  b, 
c,  d,  hab,  2cd,  and  3ef  are  the  terms. 

6.  Monomial,  Polynomial,  etc. — An  algebraic  ex- 
pression of  one  term  only  is  called  a  monomial ;  an  expres- 
sion of  two  terms  is  a  binomial ;  one  of  three  terms,  a 
trinomial.  In  general,  an  expression  of  more  than  one 
term  is  called  a  polynomial. 

7.  Factors.— Just  as  5  and  7  are  factors  of  35,  so  7 
and  a  are  factors  of  7a  ;  so  a,  b,  and  c  are  factors  of  the 
expression  abc. 

8.  Power  and  Index,  or  Exponent. — When  the 
same  factor  occurs  several  times,  as  a  x  a  x  a  x  a  x  a,  or 


DEFINITIONS.  9 

aaaaa,  we  write  it,  for  the  sake  of  shortness,  a\  This  ab, 
thus  written,  is  called  the  fifth  power  of  the  number  a, 
and  is  read  "fl-tcLtlie  fifth  power."  So,  also,  a  x  a  x  «  is 
written  a3,  and  is  read  "a  tefcfa©  third  power."  a  x  a  is 
written  a2,  and  read  "  a  squared,"  or  "  a  to  the  second." 
The  5,  3,  and  2  thus  written  are  called  exponents  or  in* 
dices. 

Definition. — An  index  or  exponent  of  a  letter  is  a 
small  number  placed  over  it  to  the  right  to  show  its  power, 
or  how  many  times  it  is  taken  as  a  factor.  When  no  ex- 
ponent is  written,  the  exponent  1  is  understood. 

Examples. — Bead  a7,  b9,  2%  34,  a2b\  b3c\  and  write 
them  with  all  their  factors. 

9.  Positive  and  Negative  Quantities. — All  terms 
or  quantities  which  have  the  plus  sign,  or  no  sign,  before 
them  are  additive,  and  are  called  positive  quantities. 

All  quantities  with  the  minus  sign  before  them  are  sub- 
tractive,  and  are  called  negative  quantities. 

10.  Like  Terms. — Like  terms  are  those  which  differ 
only  in  their  numerical  coefficients.  All  others  are  un- 
like. 

Thus,  7a,  —  8a,  and  +  5a  are  like  terms  ;  as,  also,  8a3 
and  —  6a3  ;  16a~b  and  a~b. 

11.  Like  Signs. — When  two  quantities  are  both  plus, 
or  both  minus,  they  are  said  to  have  like  signs.  When 
one  is  plus  and  the  other  minus,  they  are  said  to  have  un- 
like signs. 

Note.— The  pupil  should  now  be  practiced  in  reading  algebraic  expressions, 
and  in  writing  them  down  from  dictation. 


10  ADDITION. 

SECTION    II. 
ADDITION. 

12.  To  add  like  algebraic  quantities. 

Rule. — Add  separately  the  plus  and  minus  coefficients, 
take  the  difference  of  the  ttvo  sums,  prefix  to  this  the  sign 
of  the  greater,  and  attach  the  common  letter  or  letters. 


Examples — 1. 

(1.) 

(2.) 

(3.) 

(4.) 

2a 

hax 

-  6b 

-  lab 

6a 

4:UX 

-    3b 

-    3ab 

4a 

ax 

-      b 

-    2ab 

a 

2ax 

-  10b 

—      ab 

Ida 

12ax 

-  20b 

-  lOab 

(5.) 

(6.) 

(7.) 

(8.) 

(9.)    (10.) 

To   3a 

3a 

3a 

-la 

8a              a 

add  —  a 

±a 

~7a~ 

6a 

ha 

-10a          -b 

2a 

+  3a 

-  2a 

—    2a      a  —  b 

(11.) 

(12.) 

(13.) 

(14.) 

hx 

10a2 

lhab 

12cx2 

—  4x 

3a* 

-  lOab 

—  lex* 

+  2x 

-    4rr 

—      ab 

-    dcx2 

+  ox 

+  9«2 

+  Aab 

—      ex2 

(15.) 

(10.) 

(17.) 

5a  — 

3c 

ha 

-  3c 

ha   +  3c 

6a  - 

8c 
lc 

6a 

+  8c 

6a  -  8c 

11a  -  1 

11a 

+  he 

11a  -  he 

ADDITION. 


11 


(18.) 
6a  —  25  +  5c 
4a  +  85  -  2c 

10fl  4-  6b  4-  3c 

OWL) 

3a  —  45  —    c 
6a  4-  9*  -  7c 

-  5a  +  25  —  4c 


(19.) 

a  —  b  +  c 
a  +  b  —  c 


2a 


(20.) 

CC    —   y    -f       2 

a;  4-  ?/  4-     ^ 


2a; 


4-  2z 


(22.) 

-  21a2  -  Uab  +  20«c2  4-  30ac 
45a2  -  20ab  -  12ac*  -  \6ac 

—  da*  4-    2ab  +  25ac2  -  Mac 


Remark. —  Unlike  Terms. — When  unlike  terms  occur,  unite  them 
in  the  sum  with  their  proper  signs. 

Thus,  to  add  8c  —  5c?  to  3a-  —  b,  we  simply  write  it  3a-  —  b  +  8o  — 
5cZ ;  again,  the  sum  of  3a'—Aab  and  —  2a?  is  a-  —  4ab. 


(23.) 
Add  a  +  25  -  c  and 

a  —  6wi  4-  2c.    Thus  : 

a  4  2b  —    c 

a  -f  2c  —  6m. 

2a  +  25  -f    c  —  6w 


(24) 
Add  6a?s  —  8a;  +  a  and 
3a;  -  #  +  6.     Thus  : 

6a;2  -  Sx  +  a 

3a;  —  y  -f  6 

6ar  —  5a;  4  a  —  ?/  4-  6 


(250 
Add  3a2  +  45a;  -  c2  4-  10,    -  5a2  +  6ac  +  2c2  -  15a, 
4a2  -  3ac  -  c2  4-  21. 

3a2  4-  45a;   -  c2    +  10 

—  5a2  4-  2c2  4-  6ac  —  15a 

-  4a2  -  c2    4-  21  -  3ac 


6a'  4-  45a- 


4-  31  4-  3ac  —  15a 


12  SUBTRACTION. 

Examples — 2. 
Add  together 

1.  —  Ga,  &a,  —  loa,  3a,  —  4ft. 

2.  x  +  y  +  z,   —  x  +  y  +  z,  x  —  y  +  z,  x  +  y  —  z. 

3.  3  —  a,   —  8  —  a,  la -I,   —  a  —  1,  9  +  a. 

4.  63  -  2«&2  +  a2b,  ¥  +  3«62,  2«3  -  ab2  -  a2b. 

5.  2x*  —  W  +  3,   —  4z3  +  Gx2  -  2a;  +  7,  a*4  -  2z3  -  4a;, 
62;3  -  9a;  -  12. 

In  like  manner,  by  grouping  and  adding  like  terms, 

6.  Eeduce    the    polynomial    ax  —  Ga/b  +  Gab2  —  2ba  + 
5<24  —  3a#2  +  6#2£  +  b3  —  4«4  +  2«2#   to  its  simplest  form. 

7.  Eeduce  4«?/2  —  3^  +  Sab  +  7a#  —  Gab  +  c  +  8##2  + 
4#2  +  7##  —  c  +  7#ya  —  822  —  9ab  to  its  simplest  form. 


SECTION     III. 

SUBTRACTION. 

13.  To  subtract  +  b  from  a,  we  have  a  —  b.  To  sub- 
tract —  &  from  a,  we  can  write  for  a,  a  +  b  —  b,  as  it  is 
the  same.  Now  —  J  taken  from  a  +  b  —  b  leaves  «  +  #. 
Hence,  —  b  taken  from  a  gives  a  +  £. 

In  like  manner,  b  subtracted  from  —  a  gives  —  a  —  b, 
and  —  b  subtracted  from  —  a  gives  —  a  +  b,  —  2a  sub- 
tracted from  da  gives  +  5#. 

Hence, 

Rule. — Change  the  sign  of  every  term  in  the  subtra- 
hend, and  proceed  as  in  addition. 


SUBTRACTION. 


13 


Examples — 3. 

1. 

From 

4a 

2.  From    5x 

3. 

From      4a 

take 

a 

take      4:X 

take  —    a 

3a 

X 

ha 

4. 

From 

b 

5.  From      a 

G. 

From      5a 

take 

b 

take  —  a 

take    —  4a 

0 

2a 

9a 

7. 

From 

-4a 

8.  From  —  5b 

9. 

From  —  a 

take 

a 

take         ±b 

take         a 

—  5a 

-  9b 

—  2a 

10. 

From 

-4a 

11.  From  —  5c 

12. 

From  —  a 

take 

—  a 

-  da 

take    —  4c 

take    —  a 

—    c 

0 

13. 

From 

%  +y 

14.  From  b  —  c 

15. 

From  «  -f  bo 

take 

x  -y 

take   £  +  c 

take    a  —  ex 

2y  -  2c  Ic  +  c# 

(16.) 
From  4a  —  5b  +   7c 
take      a  —  ob  +  10c  —  4# 
^i«s.   3a— 2b—  3c  +  4z         Ans.     x  +  oy  —  1 

(18.)  (19.) 

From     a2  +  dab  -  4c2  From 

take      2a2  -  6a£  -  8c"2  take 

Ans.  —a-  +  9a  Z>  +  4c2  Ans.  —  x*+x3  +  x"  —  2x  —    2 


(17.) 
From  Sx  —  2y  +  4z  —  5 
take     7a;  -  5//  -t-  4z  —  4 


z3-4.r+   &u  -  11 
z4-5z2  +  10z  -    9 


14  MUL  TI PLICA  TION. 


Examples — 4. 

1.  From  a   take   a  —  6  —  x. 

2.  From  2a  +  3b  —  c  -  d  take   2a  -3b  +  c  -  d. 

3.  From  8a  —  I?  —  c  take   a  —  b  +  £c. 

4.  From  3«  +  2a;  —  56   take   2a  +  3x  +  46. 

5.  From  xij  +  2a;*2  +  3if   take   £«/  —  2a;'2  +  3y\ 
G.  From  4?»w  +  5m  —  Gn   take   2mn  +  m  +  w. 

7.  From   3#26  +  4«2c  —  Gc2   take   ft*2  6  —  a/c  —  8c'2. 

8.  From   \ab  —  \bc  -f  -J   take   \ab  +  foe  —  §. 

9.  From   a3  —  30«2a;  +  51<za;*2  —  84a;3   take   a  —  3oa2x 
max1  -  250a;3. 


SECTION   IV. 
M  UL  TIP LIC A  TION. 

14.  We  have  seen  that  ab  is  aaaaa,  arid  aA  is  «&«#. 
Therefore,    ar>  x  #4  is  aaaaa  x  ««««  =  a9. 

Hence,  to  multiply  powers  of  the  same  letter,  we  add  the 
exponents. 

15.  1.  -h  3a  x  +  2b  means  +  3a  added  2b  times,  or 
+  Gab. 

2.  —  3a  x  +  26  means  —  3a  added  2b  times,  or  —  Gab. 

3.  +  3a  x  —  2b  is  the  same  as  —  2b  x  3#,  or  —  Gab,  as 
above. 

4.  —  3a  x  —  26  means  —  3a  subtracted  26  times,  that 
is,  —  Gab  subtracted,  which  by  the  rule  of  subtraction 
(Art.  13)  gives  +  Gab. 


M  UL  T I  PLICA  TION.  1 5 

Hence,  summing  up,  we  have 

+  oa  x  +  2b  --  +  Gab, 

-  3a  x  -f-  2b  =  —  G«#, 
+  3«  x  —  2#  =  —  6«5, 

-  3«  x  -  26  =  +  Gab. 

Hence,  +  by  +  gives   +  ;    —  by  -  gives  +  ;  +  by  - 
gives  —  ;  and  —  by  +  gives  —  .     Therefore, 

16.  To  multiply  a  monomial  by  a  monomial, 

Rule.— 

I.  For  Coefficients  :  Common  multiplication. 
II.  For  Signs  :  Like  signs  make  +  ;   Unlike  signs,  — . 
III.   For  Exponents  :  Add  the  exponents  of  the  same  Let- 
ters. 

Examples  : 

1.  3a2  x  4«3  =  12a8. 

2.  hob  x  ±abc  =  20a*b*c. 

3.  4«'  x  —  axy  —  —  Aa'xy. 

4.  —  5x*y*z*  x  —  2x1yz1  =  lOx*  y3z*. 

5.  2ab  x  —  3cy  x  —  a'Vy  —  6a4J"cy\ 

6.  axy  x  b  ;  3ab  x  —  x ;  —  3mm  x  awi ;  —  xxf  x  —  xif . 

17.  To  multiply  a  polynomial  by  a  monomial  or  single 
term. 

Rule. — Multiply  every  term  in  the  multiplicand  by  the 
multiplier. 

Examples — 5. 

1.  5«V  -  Gabx1  +  3b*y* 


10«2foV  +  12adacV  -  G£scy 


1 G  MUL  TIPLICA  TION. 


2.   12a  -  7b.         3.   5b  -  8a.         4.   10x2  - 

-  5ax  - 

-3a\ 

9a                        -  12a                  4-x2 

Multiply 

5.   2ab  —  Aac  +  Gbd  by  —  2x. 

6.  ftc  +  2Z>6'  by  oa. 

7.   2az  +  5%  -  dcz  by  -  2^2. 

8.   oxH  —  2.Z0  +  Ax2  by  —  7#2. 

18.  To  multiply  a  polynomial  by  a  polynomial. 

Rule. — Multiply  all  the  terms  of  the  multiplicand  by 
each  term  of  the  multiplier.     Then  add  these  products. 


Ex.  1. 

Ex.  2. 

Ex.  3. 

3a  +  2b 

x  +  3 

o1  +  2z2 

ha  -  U 

x  —  2 

3a2+  a2 

15aa+10a£  rr  +  3rc  3a4+ 6aV 

-12a5-85a  -3a?  — 6  a9x* +2xA 


Prod.  15«2-2tfZ>-862  Prod.  ^2+^-6  Prod.  3«4  +  7aV  +  2a;4 


Ex.  4. 

2a    +  36    -  5c       ' 

a   -V   b     —  c 

2a2  +  oab  —  5ac 

+  2ab             +  362 

-5bc 

—  2«c 

-  Sbc  +  5c3 

2a'2  +  5afl  -  7«c  +  3b2  -  Sbc  +  5c\ 


MULTIPLICATION.  17 


Examples — 6. 

Multiply 

1.  a  +  x  by  c  +  y. 

2.  5a  +  4  by  a  —  2. 

3.  a;  —  5  by  x  +  4. 

4.  3a  -  4  by  2a  -  3, 

5.  1  —  2a  by  x  —  a2. 

6.  ac  —  b2  by  c2  —  ab. 

7.  -  11a  -  3a  by  -  10a  -  8a. 

8.  1  +  3a  +  2y  by  a  —  y. 

9.  «#  —  be  -\-  ac  by  2«  —  5. 
10.  a3  +  a2  +  x  +  1  by  a  —  1. 

Multiply 

11.  5  +  2a  +  a2  by  5  —  2a  +  x\ 

12.  a  +  4  —  £/  by  a  +  4  +  y. 

13.  3«V  +  Wy  by  3aV  -  2%. 

14.  2a3  +  4.r  +  8a  +  16  by  x  -  2. 

15.  «V  —  a*x*y  +  «2a2?/2  —  aay3  +  y*  by  #a  -f  y* 
1G.  a2  -  2a6  +  2¥  by  a:  +  2ab  +  262. 

17-  «2  +  #2  +  c2  —  «#  —  ac  —  6c  by  a  +  b  +  c. 

18.  1  -  2a  +  3aa  -  4aa  4-  5«4  by  1  +  2a  +  a\ 

19.  a2  -  5a  -  9  by  a2  -  5a  +  9. 

20.  Find  the  continued  product  of  a  —  2  by  a  —  2  by 
x  -  2. 

21.  Multiply  a9  +  2a  -  1  by  a*  —  w  +  1  and  by  a'  - 
3«  —  1,  and  subtract  the  second  product  from  the  first. 

22.  Multiply  1  -  2a  +  3a2  -  4a3  by  1  -  a  ~x\ 


18  BRACKETS  OR  PARENTHESIS. 

SECTION    V. 
BRACKETS  OR  PARENTHESIS. 

19.  Two  or  more  terms  are  sometimes  put  in  brackets 
or  a  parenthesis,  and  considered  as  a  single  term.  The 
sign  before  the  brackets  indicates  the  operation  to  be 
performed  on  all  the  terms  in  them.  If  we  remove  the 
brackets,  the  operation  must  be  performed. 

Brackets  in  Addition  and  Subtraction, 

20.  When  the  +  sign  is  before  the  brackets,  the  terms 
are  to  be  added.  Thus,  a  +  (b  +  c  —  d)=a  +  b  +  c  —  d. 
When  the  —  sign  is  before  the  brackets,  the  terms  are  to 
be  subtracted.  Thus,  a  —  (b  +  c  —  d)  =  a  —  b  —  c  +  d. 
Therefore  we  have  the 

Rule. —  When  we  remove  brackets  tvith  the  +  sign  be- 
fore them,  the  signs  of  the  terms  within  remain  unchanged. 
When  we  remove  brackets  with  the  —  sign  before  them,  we 
must  change  the  signs  of  the  terms  within. 

Ex.  1.  2a  —  b  +  (b  —  a)  =  2a  —  b  +  b  —  a  =  a. 
Ex.  2.  2  -  a  -  (2  -  2a)  =  2  -  a  —  2  +  2a  =  a. 
Ex.  3.  3  +  a  +  {-  a  —  3)  =  3  +  a  -  a  -  3  =  0. 
Ex.  4.   1  -  b  +  ¥  -  (-  1  +  2b)  -  (-  ¥  +  1) 

=  1  -  J  +  ¥  +  1-2*  4-  V  -1  =  1-  db+  2b\ 

Examples — 7. 

Simplify 

1.  6  -  (5  +  3)  +  (2  -  4)  -  (3  -  10). 

2.  a  —  b  —  c  —  (a  +  b  —  c). 


BRACKETS  OR  PARENTHESIS.  19 

3.  2x  -  dy  -  {2x  +  3y). 

4.  1  -  b  +  b%  -  (1  -  b  +  /r  -  63). 

5.  9a  +  126  +  c  —  (a  +  lb  +  c)  —  (Sa  A-  Ab). 

6.  a  —  [b  —  c  —  (d  —  c)]  =  a  —  b  +  c+  (//-  c)  —  a  —  b  + 

c  i-d  —  c=a  —  b+d. 

7.  Qa  -  [U  -  (2a  -  b)]. 

Brackets  in  Multiplication, 

21.  When  brackets  are  used  in  multiplication,  they 
mean  that  the  number  or  letter  before  or  after  the  brackets 
is  to  be  multiplied  by  all  the  terms  in  them. 

Thus,    a  (Jj  —  c )  means  a  multiplied  by  b  —  c. 

(a  +  b  +  c)x  means  a  +  b  +  c  multiplied  by  x. 

Notk.—  Instead,  of  using  brackets  to  indicate  multiplication,  we  often  use  a  line 
over  the  polynomial  as  follows  ; 

a  .  b  +c. 
This  line  is  called  a  vinculum. 

22.  When  two  pair  of  brackets  are  used  in  multiplica- 
tion, they  mean  that  all  the  terms  in  one  pair  are  to  be 
multiplied  by  the  terms  in  the  other. 

Tims,  (a  +  x)  {a  +  2x)  means  a  +  x  multiplied  by  a  +  2x. 

(a  +  b)2  means  (a  +  b)  (a  +  b),  i.  e.,  a  +  b  multiplied  by  a  +  b  ; 
(5a)*  ==  (5a,)  (5a),  i.  e.,  5a  x  5a. 

23.  Hence,  in  multiplication  we  have  the 

Rule. —  Wlien  we  remove  brackets,  we  must  first  perform 
the  multiplication  indicated. 

Thus,  a  (b  —  c)  =  ab  —  ac  ;  a  x  b  +  x  —  ab  +  ax  ; 
(a  +  x)  (a  +  2x)  =  a{a  +  2x)  +  x{a  +  2x)  =  a2  +■  2ax  +  ax  +  2x°-  - 

a2  +  3ax  +  2x'2  ; 
5  (x  _  2)  -  6  (x  -  3)  =  5x  -  10  -  6z  +  18  =  -  x  +  8. 


20  MULTIPLICA TION  B  Y  INSPECTION, 

Examples — 8. 
Simplify 

1.  6  (x  -  5)  +  3  (x  -  4)  -  5  {x  -  2). 

2.  a  (b  —  6')  —  b  (a  —  c)  +  c  (a  —  b). 

3.  (x  -  5)  (x  +  5)  -  G  (x*  -  25). 

4.  (a  +  b  +  c)  x  —  {a  +  b  —  c)  x. 

5.  (3x  -  2)  (3<e  +  2)  -  (dxy. 


SECTION    VI. 
MULTIPLICATION  BY  INSPECTION. 

24.  Some  polynomials  can  be  multiplied  readily  by  in- 
spection— that  is,  without  putting  the  quantities  one  under 
the  other  and  proceeding  by  the  regular  rule — if  we  first 
learn  certain  forms  and  rules.  The  following  four  cases 
arc  of  most  frequent  application. 

The  Square  of  the  Sum  of  Two  Quantities. 

25.  (a  +  b)\ 


Operation. 

a  +  b 

a  +  b 

a2  +  ab 

+  ab  +  b* 

a'  +  2ab  +  V 

cr  +  2ab  +  If 

.     .     .     .     (A) 

or  (a  +  by 

This  is  called  a  formula,  and  expresses  in  algebraic  lan- 
guage the  following 

Rule.-—  The  square  of  the  sum  of  two  quantities  is  the 
square  of  the  first,  phis  twice  the  product  of  the  first  by  the 
second,  plus  the  square  of  the  second. 


MULTIPLICATION  BY  INSPECTION.  21 

Ex.   1.   (x  +  5)2  =  x2  +  10ic  +  25. 

Ex.  2.   (da  +  2b)2  =  (3a)2  +  2  x  3a  x  2b  +  (26)2  = 
9a2  +  12«6  +  U2. 

The  Square  of  the  Difference  of  Two  Quantities. 

26.  (a  —  b)2.  Operation.       a  —  b 

a  —  b 


a2  —  «Z> 

-  ab  +  62 
«2  -  2«6  +  62 
or  (a  -  b)2  =  a2  -  2ab  +  b2.     .     .     .     (B), 

which  expresses  the  Rule  : — TJie  square  of  the  difference 
of  two  quantities  is  the  square  of  the  first,  minus  twice  the 
product  of  the  first  by  the  second,  plus  the  square  of  the 
second. 

Ex.   1.   (x  -  5)2  =  x2  -  10s  +  25. 
Ex.  2.   (da  -  2b)2  =,  (3a)2  -  2  x  3a  x  2b  +  (2b)2  =s 
9a2  -  12a5  4-  462. 

T/*e  Swm  o/  Two  Quantities  Multiplied  by  their 
Difference. 

27.  (a  +  b)  (a  —  b).  Operation,     a  +  b 

a  —  b 


a2  +  ab 

-  ab  -  b- 
a2  ^~b2 

or  (a  +  b)(a-b)  =  a2  -  b2   .     .     .     .(C), 

which  expresses  the  Rule  : — Tlie  sum  of  two  quantities 
multiplied  by  their  difference  is  the  square  of  the  first 
minus  the  square  of  the  second. 


22  MULTIPLICATION  BY  INSPECTION. 

Ex.  1.  (as  +  5)  (x  -5)=  x*  -  52  =  .r  -  25. 

Ex.  2.  (2a  +  3b)  (2a  -  3b)  =  (2a)"-  (3b)'  =  4«2-  W 

Ex.  3.  (46  +  3)  (46  -  3)  =  (U)2  -  33  =  1G62  -  9. 


The  Product  ofx  +  or  —  one  Number  by  x  +  or 
another  Number. 


28.  (x  +  a)  (as  +  b). 


Operation,     x  +  a 
x  +  b 


x  +  aas 

+  bx  +  ab 
x"  +  («  +  6)a  +  a# 


or 


(as  +  a)  (x  4-  6)  =  x2  +  (a  +  J)as  +  «6 


Also, 

(as  —  a)  (x  —  b)  =  x*  —  (a  +  b)x  +  ab 

(x  +  a)  (x  —  b)  —  x1  +  (a  —  b)x  —  ab  J 


(D). 


Hence,  the  Rule: — The  product  of  xplus  or  minus  a 
number,  by  x  plus  or  minus  another  number,  is  x2  plus  or 
minus  the  (algebraic)  sum  of  the  numbers  multiplied  by 
x,  plus  or  minus  the  product  of  the  numbers,  according  to 
the  sign. 


Ex.  1.   (x  +  2)  (x  +  3; 

x1  +  5x  4-  G. 


3) 


Xs  +  (2  +  3)  x  +  2  x  3  = 
as3  -  (2  +  3)  as  +  (-  2  x 


Ex.  2.   (as  -  2)  (x 
-  3)  =  as2  -  5as  +  6. 

Ex.  3.   (x  +  3)  (as  -  2)  =  x1  +  (3  -  2)  x  f  3  x  -  2  = 

as2  -4-  ^  —  G. 


DIVISION. 
Ex.  4.    (x  -  3)  (x  +  3)  =  s9  +  (2  -  3)»  4-  2  x 

:  .T2    -  Z   -   6. 


Examoles — 9. 


Write  the  squares  of 


1.  m  +  n  ;   m  —  n. 

2.  m  —  2n  ;   3#  +  46. 

3.  4«  -  6  ;   5«  -  36. 

4.  7x  —  4y  ;   #2  Jr  3##. 

5.  1  +  2«2 ;   3:r  -  2a2. 

Write  the  products  of 

G.  (a  +  2b)  (a  -  26)  ;    (2a  +  6)  (2«  -  b). 

7.  (z  +  3#)  (x  -  3y)  ;    (3a  +  4c)  (3«  -  4c). 

8.  (8x  +  9y)  (8x  -  9y)  ;    (za  +  if)  (.*"  -  ?/2). 

9.  (3ax  +  6)  (3az  —  b)  ;    (ma;  +  2«#)  (mx  —  2a?/). 
10.  (4x2  +  1)  (4za  -  1)  ;    (x*  -  4)  (x*  +  4). 

Write  the  products  of 

11.  (x  +  5)  (x  +  3)  ;  (x  -  6)  (x  -  4). 

12.  (x  -  G)  (z  +  2)  ;  (x  -  8)  (x  +  10). 

13.  {x  +  7)  (a  -  0)  ;  (a  +  6)  (a  -  c). 


SECTION     VII. 

DIVISION. 

29.    +  a  x  +  b  =  -h  ab  ;   hence    +  fl5 
fa  x  —  5  =  —  ab ;   hence    —  ab 


+  a  =  +  b. 
+  a  —  —  b. 
a  x  +  b  —  —  a3  ;   hence    —  ab  -i a  =  -h  b. 


a  x 


b  =  +  ab  ;   hence    +  ab 


a  —  —  b. 


24  DIVISION. 

Hence,  for  signs  in  division,  we  have  the 

Rule. — Like  signs  give  +,  and  unlike  signs  give  —  . 

a6 

30.  «4  x  a1  =  «6 ;   hence,    —  =  a\ 

a 

Therefore,  to  divide  two  powers  of  the  same  letter,  we 
subtract  the  exponent  of  the  divisor  from  that  of  the  divi- 
dend. 

31.  Hence,  to  divide  one  monomial,  or  single  term,  by 
another,  we  have  the 

Rule. — Divide  the  coefficients,  observing  the  rule  of  the 
signs.  To  this  quotient  annex  the  letters,  subtracting  the 
exponents  of  the  like  letters. 

Ex.   1.    -  Cja'F  ~  3a'b  =  -  2a2b. 

Ex.   2.   Wb  ~  ±ab  =  2a. 

Ex.  3.   -  lOcrbc  -f-  5a  =  —  2abc. 

Note.— It  is  plain  that  if  the  coefficient  of  the  dividend  is  not  exactly  divisible 
by  the  coefficient  of  the  divisor,  or  if  a  letter  enters  the  divisor  which  is  not  in  the 
dividend,  or  if  a  letter  in  the  divisor  has  an  exponent  greater  than  the  exponent 
of  the  same  letter  in  the  dividend,  the  division  is  impossible,  since  in  all  these 
cases  the  divisor  has  factors  not  contained  in  the  dividend.  In  such  case,  the 
operation  is  expressed  after  the  manner  of  a  fraction  in  arithmetic. 

Examples — 10. 

Divide 

1.  8.?;  by  8  ;  8a  by  —  a  ;  abc  by  —  a  ;   —  axy  by  y. 

2.  12xy  by  3x  ;   —  Saxy  by  +  Sax. 

3.  -  15ab2c  by  -  Sab  ;   -  20«  W  by  5abc. 

4.  —  4cV£V  by  —  d'bc2 ;  J4«V?/  by  lay. 


DIVISION. 


25 


32.  To  divide  a  polynomial  by  a  monomial. 

Rule. — Divide  every  term  in  the  dividend  by  the  di- 


visor. 
Ex.   1. 

Ex.  2. 
Divide 


Sx )  40za  -  2±ax  ; 
Quotient,       ox  —  3a 

ox  )  12.r3  -  21ax'i  +  3a*x 
Quotient,  4uv*  —  lax  +  a* 

Examples — 11. 

1.  2ab  +  3ac  —  kad  by   a. 

2.  ax  -f  bx2  —  cxy   by    —  x. 

3.  6«V  -  Sabx  +  2axs   by    -  2«z. 

4.  Uc  +  35a£c2  -  10£V   by   5bc. 


Division  of  Poly  nominate. 

33.  Definition. — The  terms  of  a  polynomial  are  said 
to  be  arranged  according  to  a  given  letter,  when  beginning 
with  the  highest  power  of  that  letter  we  go  regularly  down 
to  the  lowest,  or  when  we  begin  with  the  lowest  and  go  up 
to  the  highest. 

Thus,  4c5  —  3x*  +  2x-  —  1   is  arranged  with  reference  to  x. 

34.  To  divide  one  polynomial  by  another. 

Rule. — 1.  Set  down  the  divisor  and  dividend  as  in 
long  division  in  arithmetic,  taking  care  to  arrange  them 
both  according  to  the  same  letter. 

2.  Divide  the  first  term  in  the  dividend  by  the  first  term 
of  the  divisor,  for  the  first  term  of  the  quotient.  Tfien 
2 


26  DIVISION. 

multiply  the  divisor  by  the  first  term  of  the  quotient,  and 
subtract  the  product  from  the  dividend.  Divide  the  first 
term  of  the  remainder  by  the  first  term  of  the  divisor,  and 
proceed  as  before,  continuing  the  process  with  the  terms 
that  remain. 

Ex.  1.  Divide  x~  +  kax  +  3rr  by  x  +  3a. 

x  +  3a )  x"  +  &az  +  3«2  (x  4-  a,  quotient. 
x'2  4-  3«£ 


ax  +  3«'2 

a:c  4-  oa2 

0 

Ex.  2.  a  —  b)a"  —  2ab  4-  &2  ( «  —  b,  quotient 

a'2  —  ab 


-ab  +  b'2 

-ab  +  V 

0 

Ex.  3.  Ix- 3 )  7z8  -  24za  4-  58a;  -  21  ( z2  -  Bx  4-  7,  quotient. 

7;<:3-    3.1'2 


-2Lr  +  58.£ 
— 2Lr4-   9.?; 


Divide 


4-49.C-21 

+  49a; -21 

0 

Examples — 12. 


1.  a?  4-  6x  +  6  by  ay'+  3. 

2.  8a2  4-17^  +  9  bv  8a  4-  9. 


DIVISION.  27 

3.  21a2  +  ax  -  2.r  by  7«  -  2x. 

4.  x2  -  ldx  +  40  by  x  -  5. 

5.  x2  +  y2  +  z1  +  %xy  +  %zz  4-  2yz  by  x  -f  ?/  +2. 
G.  a2  —  b2  —  c2  —  2bc  by  a  —  b  —  c. 

7.  a2  —  a  —  30  by  «  +  5. 

8.  a4  +  a2b2  +  5*  by  a2  +  «&  +  &2. 

9.  «3  —  3«26  +  oab2  —  tf  by  a2  —  2ab  +  £2. 

10.  x*  +  7.r  +  2.r  +  15  by  x2  -  x  +  5. 

11.  .ry  -f  2.*r  —  3?/2  —  4:y z  —  £2  —  22  by  2x  +  3y  +  z. 

12.  15«4  +  10a3 x  +  4aV  +  6rw3-  3.i-4  by  3«2-  x2  +  2«£. 

13.  ab  +  2«2  —  ob2  —  4fic  —  ac  —  c2  by  2a  +  ob  +  c. 

14.  3z4  +  llr5  +  Ox  +  2  by  a2  +  5.«  +  1. 

15.  3x*  +  kabx2  -  6a2b2x  -  4aW  by  2tf£  +  x. 

35.  Remark. — Incomplete  Polynomials. — Sometimes  some  of  the 
powers  of  the  letter  are  wanting  in  the  dividend  or  divisor,  or  in  both. 
In  such  case  it  is  best  for  beginners  to  leave  a  space  for  the  wanting 
terms. 

Ex.  16.  Divide  ab  -  32  by  a  ~  2. 

a -2)  a"  -  32  (a*  +  2a'  +  la2  +  8a  +  16. 

ab  -  2«4 


+  2a4 

-32 

+  2a*  -  4«3 

+  4a3 

-  32 

+  hC  -  8a2 

+  Sa2 

-32 

+  8a2  -  lGrt 

+  I60 

-  32 

+  16a 

-  32 

28  FACTORING. 

Ex.  17.  Divide  Go;4  -  96  by  2a;3  +  4a;2  +  8a;  +  16. 

2x3  +  4a;2  +  Sx  +  16 )  Gx*  -  96  (3a?  -  6 

6.t4  +  12a;3  +  24a:2  +  48a: 


-  12a;3  -  24a;2  -  48a;  -  96 

-  12a;3  -  24a;2  -  48a;  -  96 


0 

Divide 

18.  9rt262  -  16a;2  by  3ab  +  4a;. 

19.  8a;3  -  2W  by  2x  -  3a. 

20.  16«4  -  81  by  2a  -  3. 


SECTION    VIII. 
FACTORING. 

36.  By  reversing  the  formulas  in  Section  VI., 

we  may  determine,  by  inspection,  the  factors  of  certain 
polynomials. 

We  have  seen  that  (x  +  a)  (x  —  a)  —  x*  —  a\  (Art.  27.) 

Hence,  the  factors  of  a;2  —  a2  are  x  -\-  a  and  x  —  a. 
Thus,  also,  the  factors  of  a;2  —  25  are  x  +  5  and  x  —  5  ;  of 
9a;2  -  4  are  3a;  -f  2  and  3a;  —  2. 

37.  (x  +  a)-  =  x2  +  2ax  +  a' ;  hence,  the  factors  of 
x~  +  %ax  +  «2  are  a;  +  #  and  x  +  a  ;  of  ar  +  10a;  +  25 
are  x  +  5  and  a;  +  5.     (Art,  25.) 

38.  (x  —  af  —  x*  —  2ax  +  a2 ;  hence,  the  factors  of  x"'  — 
2«a;  -f-  a"2  are  x  —  a  and  a;  —  a.    (Art.  26.) 

Ex.  The  factors  of  a;2  —  8a;  +  1 6  are  a;  —  4  and  x  —  4. 


FACTORING.  29 

39.  We  have  seen  (x  +  a)  (x  +  b)  —  x-  +  (a  +  #)#  +  «& ; 
hence,  the  factors  of  a;1  +  (a  +  #)#  +  «&  are  a;  +  a  and 
a;  +  J.     (Art.  28.) 

Ex.  The  factors  of  a;2  +  ox  +  6  are  «  4-  3  and  a;  +  2. 

40.  (a;  —  a)  (x  —  5)  =  a:2  —  («  +  #)#  +  a&  ;  hence,  the 
factors  of  x2  —  (a  +  #)a;  +  ab  are  x  —  a  and  x  —  b. 

Ex.  The  factors  of  or2  —  7a;  +  10  are  x  —  5  and  a;  —  2. 

41.  (a;  +  a)  (x  —  b)  =  a;9  +  (a  —  #)a;  —  rt&  ;  hence,  the 
factors  of  x1  +  (rt  —  b)x  —  ab  are  a;  +  a  and  a:  —  b. 

Ex.  1.   The  factors  of  x*  +  f>x  —  6  are  a:  +  6  and  a:  —  1. 

Ex.  2.  The  factors  of  a;9  —  5x  —  6  are  a;  —  6  and  a?  4-  1. 

Examples — 13. 
Find  the  factors  of 

1.  x2  +  4a:  +  4  ;  2?9  -  12a;  ■+  36. 

2.  4a:2  -  12a;  +  9  ;  or  +  2«c  +  c\ 

3.  a;'3  —  4  ;  4a2  —  4«c  +  r. 

4.  4a;2  -  9  ;  IQcrV  -  9c2. 

5.  16ft2  -  G4  ;  9mV  -  25. 

6.  a;4  -  a4  ;  36#V  -  25&y  ;  1  -  4a;2. 

7.  a;2  +  9a;  +  20  ;  x2  +  4a;  +  3. 

8.  a;2  -  Gx  +  8  ;  a;2  +  a;  -  6. 

9.  s2  —  a?  —  6  ;  x'  -  8a;  -  20. 

10.  a:2  -  Gx  -  7  ;  ar  +  6a;  -  7. 

11.  x2  -  x  -  72  ;  a-'  -  2a;  -  99. 

12.  x-  —  12ax  +  32«2 ;  «V  —  4dxbx  +  4S2. 

13.  x2  -ex  —  110c-  ;  xn-  -  %lcx  +  110c2. 


30  GREATEST  COMMON  DIVISOR. 

SECTION    IX. 
GREATEST  COMMON  1)1 VI SOB. 

42.  To  find  the  greatest  common  divisor  of  two  mono- 
mials. 

Rule. — Find  the  G.  C.  I),  of  the  coefficients  by  the  rule 
in  arithmetic,  and  of  the  letters  separately  by  inspection, 
and  multiply  the  results. 

Ex.  1.  Find  the  G.  C.  D.  of  l§abVc  and  45«45V. 

The  G.  C.  D.  of  18  and  45  is  9  ;  of  a'  and  ab  is  a*  ;  of  ¥  and  b* 
is  b-  ;  of  c  and  c4  is  c. 

Hence,  the  required  Gr.  C.  D.  is  da^b-c. 

Ex.  2.  The  G.  0.  D.  of  45aV#  and  GOafy9  is  IZx'y. 
Ex.  3.  The  G.  C.  D.  of  54&W  and  72¥cixi  is  18JscV. 

43.  To  find  the  greatest  common  divisor  of  two  poly- 
nomials. 

Rule.— Proceed  by  the  rule  for  numbers  in  arith- 
metic :  Divide  the  greater  polynomial  by  the  less,  then 
divide  the  less  by  the  remainder,  and  then  the  first  re- 
mainder by  the  second  remainder,  and  so  on  till  there  be  no 
remainder.  The  last  divisor  will  be  the  Greatest  Common 
Divisor. 

Note.— We  may  take  out  a  numerical  factor  from  any  divisor,  or  multiply  any 
dividend  by  a  numerical  factor,  if  necessary,  to  make  the  division  possible.  Any 
factor  common  to  all  the  terms  of  both  polynomials  is  a  part  of  the  G.  C.  D. 


GREATEST  COMMON  DIVISOR.  31 

Ex.  1.  Find  the  G.  0.  D.  of  Gar  -  llz  +  4  and  2z2  - 

bx  +  2. 

2xi-5x  +  2)Qx'i-llx  +  4:(3 
6rc2  -  15a;  +  6 


4x 

-2 

or 

2  (2a; 

-1). 

2x- 

-l)2a;2 

2a;2 

-5a*  +  2(a?- 

—     X 

-2 

-4r  +  2 

-4a;  +  2 

Hence,  2a; 

-lis 

the  G.  C. 

D. 

0 

Ex.  2. 

Find 

the  G.  C. 

D.   of 

8x2 

-f  7z 

—  1  and 

Ox' 

+ 

?#  +  1. 

7a;  -1 

&K2  +     7x 
4 

+  1 

8a;5 + 

)  24c2  +  28a; 

+  4(3 

24k2  +  21a; 

-3 

7x  +  7 
or  7(a; +1). 

a;  4- l)8a;*  +  7a;-  1  (8a;-  1 
82;*  +  82; 


-  a;-l 

-  x  -1 


0  Hence,  a;  +  1  is  the  G.  C.  D. 


Examples — 1 4. 

Find- the  G.  C.  D. 

1.  Of  64«W    and    48«4^. 

2.  Of  78«-^3       and     52«s\ 


32  LEAST  COMMON  MULTIPLE. 

3.  Of  9GcV#2  and        108cy. 

4.  Of  x"  +  2x  +  1  and        x*  -  bx  -  6. 

5.  Of  x1  —  4#  +  4  and        a;2  -  5a:  +  6. 

6.  Of  a?  +  8z  -  9  and        x"  +  17.t  +  72. 

7.  Of  x2  —  %x  +  1  and         3a9  -  5#  +  3. 

8.  Of  3ar2  —  Kto  +    8  and         O.r-  -  5#  -  4. 

9.  Of  3x°-  -  20#  +  32  and  15x2  —  64z  +  16. 


SECTION    X. 

LEAST  COMMON  MULTIPLE. 

44.  To  find  the  least  common  multiple  of  two  or  more 
monomials. 

Rule. — Find  the  L.  C.  M.  of  the  numerical  coefficients 
by  the  rule  in  arithmetic,  and  of  the  letters  separately  by 
inspection,  and  multiply  the  results. 

Ex.  1.  Find  the  L.  C.  M.  of  4«V  and  6ax\ 


Result, 


2)4,  6 

a  )  ax*,  a-x" 

2,  3 
12. 

x )    x%,    ax" 
x )    x-,    ax 

x,     a 

Result,     axxxa  —  atx3, 

Hence,  L.  C.  M.  =  12a*a3. 

Note.— The  rule  in  arithmetic  applies  to  polynomials  also. 


REDUCTION  OF  FRACTIONS  TO  LOWEST  TERMS.      33 

Ex.  2.  Find  the  L.  C.  M.  of  gf  -  4  and  o&  -  2x  +  1. 

x  —  2 )  x'2  —  4         x"2  —  2x  +  1 


x   +  2 


L.  C.  M.  =  (a?  -  2)  (a  +  2)  (rz  -  2) 

(x  -  2)(z2  -  4)  =  re"  -  2z2  -  Ax  +  8. 

Examples — 15. 

Find  the  L.  C.  M.  of 

1.  2a,  V2ab,  and  Sab. 

2.  a*,  b\  and  2bc. 

3.  16«2,  12a3,  and  30«4. 

4.  a#,  «c2,  arc,  and  5c. 

5.  8z4,  IWy,  and  12^y. 

6.  7a5,  42«%  and  63«6. 

7.  18a£3  and  12«3Z>. 

8.  ax  +  #7/  and  ax  —  #?/. 

9.  2(«  +  b)  and  6(a9  -  b2). 

10.  a;2  -  2a  +  1  and  x*  —  3x  +  2. 


SECTION    XL 

REDUCTION  OF   FRACTIONS   TO  LOWEST 
TERMS. 

45.  To  reduce  a  fraction  to  its  lowest  terms. 

Rule. — The  same  as  in  arithmetic : — Divide  the  numer- 
ator and  denominator  of  the  fraction  by  their  greatest 
common  divisor. 
2* 


34      REDUCTION  OF  FRACTIONS  TO  LOWEST  TERMS. 

Ex.  1.  Reduce    =-=-=-*  to  its  lowest  terms. 
15a2x 

Cancelling  out  like  factors,  we  have 
4 


Ex.  2.  Reduce    ^-^ ^tt    to  its  lowest  terms. 

3ab  —  2o~ 


2ab  -    b°-       (2a  -    b)b         2a 


dab  -  2¥  ~  (3a  -  2b)  b         3a  -  2b 

X'  —  9 
Ex.  3.  Reduce    — ^  . 

x-  +  6x  +  9 

x*  -9  (x  —  3)  (x  +  3)      x  -  3 


x-  +  Qx  +  9  ~         (z  +  3)9        _  x  +  3 

Examples — 16. 
Reduce  to  their  lowest  terms 


ax        abc 
bx '      bed  * 


3a  -  35        a3  -  afl 
36  "  '  ab 


25a&V  «5^_ 

15a V  '  2a£2c4' 

x*  +  xy  4:a*b  —  5W 
x*  —  xy'  %i)abc 


REDUCTION  TO  COMMON  DENOMINATOR.  35 

x*  —  y-         x*  —  2x  -f  1 


5. 


6. 


x*  +  xy  '      x~  —  3x  +  2  ' 
x-  —  a"  x-  —  4 


x'  +  2ax  +  «-' '     ar  +  4a;  +  4 


SECTION     XII. 

REDUCTION  OF  FRACTIONS   TO   A    COMMON 
DENOMINA  TOR. 

46.  To  reduce  fractions  to  a  common  denominator. 

Rule. — The  same  as  in  arithmetic  : — Multiply  each 
numerator  by  all  the  denominators,  except  its  own,  for 
new  numerators,  and  the  denominators  together  for  the 
common  denominator.     Or, 

Multiply  each  numerator  by  the  quotient  resulting  from 
dividi?ig  the  least  common  multiple  of  the  denominators  by 
its  denominator,  and  ivrite  the  L.  G.  M.  of  the  denomina- 
tors for  the  common  denominator. 

XXX 

Ex.  1.    Reduce    -r- ,  — ,  —    to  a    common   denomi- 
4        o        o 

nator. 

The  least  common  denominator  is  24.     Hence  the  result, 

6x       8x      dx 
24'     24'     24* 

Ex.  2.  Reduce  ^- ,  0j  and  — —  to  a  common  de- 
nominator. 


36  REDUCTION  TO  COMMON  DENOMINATOR, 

36ic3  is  the  L.  C.  M.  of  the  denominators.     Hence  the  result, 

6acx        4:bc        abx*- 
36^'     36a? '     36a?  ° 

Examples — 17. 

Seduce  to  a  common  denominator 


X  X  -,         X 

L  %'    T'    and    15 


a¥       b  +  a  ,      _a?_ 

Z'  Tex'     "W  16gs° 


3.  a     and 

a  —  x 


4.  — ,     — ,     and 


B    5ar  +  4  ,      10a:  +  17 

5.  --^-      and       -_-, 


n      5x  —  1  -.        27  +    2 

6.  — = and     -£«— 

4a  26a 


6a  ,        ha 

7.  r    and 


a?  —  4  a;  —  3 


4a           3a  ,        6a 

8. , ,    and    -= — -. 

x—Z'x  +  Z'  x*  —  4 


5  6x  4 

a;  '    a;  —  1 '    x  +  1 


ADDITION  OF  FRACTIONS.  37 

SECTION    XIII. 
ADDITION   OF   FB  ACTIONS. 

47.  To  add  fractions. 

Rule. — The  same  as  in  arithmetic : — Reduce  the  frac- 
tions, if  necessary,  to  a  common  denominator,  then  add  the 
numerators  and  tvrite  their  sum  over  the  common  denomi- 
nator. 

Ex.  1.  Add  -,  -r,  and  -. 

O        4:  0 

The  L.  C.  M.  of  the  denominators  is  60. 

20x        15a;  12a;    _ 

TO"  +   "60    +      60     — 

20a;  +  15.r  +  12a;         47a; 


60  _     60 


5  10 

Ex.2.  Simplify  -^  + -^ . 

The  common  denominator  is  x2  —  1. 

5(a+l)       10(s-l) 
Hence,  we  have        tf  —  \    +  ~~&  —  1 


5.r  +  5  +  lOz  -  10       15a; 


x-  -  1  x-  -  1 


Ex.  3.   Simplify  ^  +  y  +  26. 

The  L.  C.  M.  of  the  denominators  is  12. 

_  9a      105      245 

Hence,  we  have    ^  +  -^-  -h  -^  = 


9a  +  105  +  245  _  9a  +  345 
12  "       12 


38  ADDITION  OF  FRACTIONS. 

Examples — 18. 


Add 


1.   -,     -,    and     -, 
a       a  a 


2.  and     — 

x  3a; 


n      X  X  ,       X 

3.    $,     cp    and     - 


4.   ^-,    — ,    and    j- 
%a      da  4a 


OX  OX  -,        i  X 

5-  T'    T'  and    30 


„      a  a  ,       « 

3c        66c  86c 


„    a  +  b         n     a  —  6 
7.    -y-    and     -j- 


.     3a;  +  1  ,     4iB  -  5 

8.    — s and 


24 

4a;  -  5  ,     3a;  -  4 

9.  a,    — £ — ,     and    — g — 

n     J  —  Grt 

10.   2rt     and     — ^ . 


11.   a    and     -  . 
c 


SUBTRACTION   OF  FRACTIONS. 
12.  x    and      — ~ . 


_,n    a  —  x       x  —  y         ,      y  —  a 

13.  ,  — ,   and     * . 

5  10  20 

14.  Simplify  T  H — r  +  ^ T 

1     -    x  —  1       a;  +  1       a?  —  1 


15.   Simplify  8  +  ^— — 


16.  Simplify  — „ —  +  — ^ — 


SECTION     XIV. 

SUBTRACTION   OF  FRACTIONS. 

48.  To  subtract  fractions. 

Rule. — The  same  as  in  arithmetic: — Reduce  the  frac- 
tions, if  necessary,  to  a  common  denominator.  Subtract  the 
numerator  of  the  subtrahend  from  that  of  the  minuend 
and  place  the  result  over  the  common  denominator. 

Ex.  1.    From  —     take       ^ . 
The  common  denominator  is  GO. 

8a!        7x         S6x         35#        36a;  -  35a  _   x 


Hence, 


5         12  60         "CO  60  60 


40  SUBTRACTION  OF  FRACTIONS. 

Ex.  2.  From  a    take    — = . 

4 

We  hava 

4a       4a  —  b      4a  —  (4a  —  b)      4a  —  4a  +  b        b 


Note.— The  -  before  -^-r —  belongs  to  the  whole  numerator,  and  hence  4«  —  b 
must  be  put  in  brackets  with  —  before  it  when  the  common  denominator  is  written. 


Examples — 19. 
i      -n  5X      ,    ,  9X 

1.  From   -y     take    yj 


Sx 
2.  From    x    take     -^ . 


3.  From   x    take    — ^ — 


Bx  +  9      ,  ,       4.^  +  3 
4.  From   — ^—     take    — g — 


5.  From   — 5 —    take     — ? — 


6.  From   T    take 


«  -f  #  a  —  b 


2       5  3       4 

7.   From   -  +  -     take     -  +  -- 


8.  From   -    — r    take    — —^ 
x  +  1  a  +  * 


9.  From    -    take 


x-2  x*—±' 


10.  From         —     take 

x  +  y  x 

111?  a  +  b         .        «  —  6 

11.  Jbrom         — ^     take  — T . 

«  —  6»  «  +  o 

12.  Prom   -  +  3-5     take     -gj- 

13.  Simplify    ac . 


14.   Simplify -  h — - 

1     "    x  +  1       z  —  1         ar-1 


SECTION     XV. 
MULTIPLICATION  OF  FRACTIONS. 

49.  To  multiply  fractions. 

Rule. — The  same  as  in  the  arithmetic  : — Multiply  the 
numerators  together  and  the  denominators  together,  after 
cancelling  out  factors  whicli  are  the  same  in  the  numera- 
tors and  denominators. 


Ex.  1.     v-  x  5  =  t-  . 

o  b 


42  MULTIPLICATION  OF  FRACTIONS. 


a       c  _  a  x  c  _  ac 
Ex.  2.     g-  x  j  =  y7^  -Yd* 


Examples — 20. 


1.  Multiply    J-j-|    by    10. 


2.  Multiply   — by    2a. 


Qr  K 

3.   Multiply       1(,         by     9(3. 


4.   Multiply   --     by     — 


O  A  Q 

5.   Multiply    -^—       by     T 


6.  Multiply   T-    by    w 


7.  Find  the  continued  product  of 
ma?    -  27ml?  oa 


Multiply   1  +  —     by     1  --  — 


9.   Multiply    r^  +  j-?-     by     T 


DIVISION   OF  FRACTIONS.  43 

10.  Multiply   1 ~     by     1  +  ~— . 

rj  a  +  I      J  1  —  a 


-.i     Tv/r  14.-  i      x~  —  %x  +    4  z2  —  7a;  +  10 

11.  Multiply  ^^^—  r5     by    g8_3^+    2 


SECTION     XVI. 
DIVISION  OF  FBACTIONS. 

50.  To  divide  fractions. 

Rule. — Tlie  same  as  in  arithmetic  : — Invert  the  divi- 
sor and  proceed  as  in  multiplication,  cancelling  when  pos- 
sible. 

Examples — 21. 


1.  Divide by     a. 

cy        J 


2.  Divide  "—    by    b. 
by       J 


3.  Divide  °~    bv    7. 

4 


4.  Divide  -.-    by     bx. 


5.  Divide  Sab    by     — - 

J     hm 


U  SUBSTITUTION. 


n    .p..   .  ,     2a  —  6ac    ,        _ 
6.  Divide by    2a. 


7.  Divide  -  —     by     —  r- 
a;  ^  ox 


_    _.  .,         «":r?/      .  ay 

8.  Dmde  -  ^JL    „y      _  ^ 


9.  Divide  2#  H bv    £ 

#       *  a; 


1n    -p..  .,     10(z  +  ?/)     .         2{x  +  y) 
10.  Divide  -£ ±1    by    -^ ^~ 


11.  Divide  5 —    by     — 7—  . 


12.   Divide by     = K —  , 

x  —  bx  +  6       J  x1  —  9 


SECTION    XVII. 


FINDING  NUMERICAL  VALUES  BY  SUBSTI- 
TUTION. 

51.  We  find  a  numerical  value  for  an  alge- 
braic expression  by  substituting  numbers  for  the  letters 
in  the  expression  and  performing  the  operations  indicated 
by  the  signs. 

Ex.  1.  If  a  =  6  and  b  =  2,  then  a  —  J  =?  6  —  2  =  I J 

«a  -  £2  =  36  -  4  =  32. 


SUBSTITUTION.  45 

Ex.  2.  If    a  —  2,    b  =  3,    c  =  4,    then   2c3  -  ftfo  = 
2  x  64  -  2  x  3  x  4  =  128  —  24  =  104. 

Examples — 22. 

Find  the  numerical  values  of  the  following  expressions 
when  a  =  1,  5  =  2,  e  =  3,  d  —  \,  /  =  5  : 

1.  ft  -  5. 

2.  -  ft  -  5. 

3.  —  ft  —  #  —  c.  ^ 

4.  a  —  b  —  c  —  d. 

5.  a  —  b  —  c  +  d  —  f. 

6.  c  +  ft1  +  5  —  ft  — /• 

7.  ftfo  ;     r/^ctZ ;     ab  +  ac  —  be  —  2bd. 

8.  5ft2  +  62  +  c-  ;     3ft'J  +  52  +'  c1  -  6«T  -  5/. 

Find  the  numerical  values  of  the  following  expressions 
when  ft  =  10,  J  =  4,  <?  =  3,  d  =  2,  0  =  0,  /==  1. 

.ft       ft        ftfo 

10.  4a  +  22>  -  (3c  -  2/). 

11.  a  -  (i  -  e  -  d) ;    2a  +  (b  -  c  -  df). 

C  —  ft  ft  —  0 


n    a       a        ft       ,   ;       35d 
'6        c       5c  '  4ftc 


14.  («  +  by  -a"-  V  ;    (ft  4-  5)2  -  (ft  -  b)\ 


46  SIMPLE  EQUATIONS. 

16  eA-cl-fl 

10'  a"        d*        c3  ' 

16.  a(ft  +  c)  4-  b(a  -  c)  -  c(f-  a) ;  (a  -  2)  (b  -  c] 


17- 5  ~  -^- ;  t  +  y  "  6 


18.  4 


3 


8(«  -  1)        8(«  +  1) 


SECTION     XVIII. 
SIMPLE  EQUATIONS. 

52.  A  principal  object  of  algebra,  as  of  arith- 
metic, is  to  find  from  numbers  which  are  known  other 
numbers  which  arc  unknown. 

53.  To  do  this,  we  put  a  letter  for  the  unknown 
number,  then  make  an  equality  from  the  given  conditions  ; 
this  we  call  an  equation,  and  from  it  find  the  value  of  the 
unknown  letter. 

For  example  :  If  x  4-  4  =  9,   then  x  =  9  —  4,  or  x  =  5. 

54.  Note.— All  expressions  with  the  sign  =  between  them  are  not, 
however,  equations  in  the  above  sense,  but  are  sometimes  identities, 
that  is,  equalities  in  which  both  sides  arc  the  same. 

Thus,  3+5  =  8  is  an  identity :  x  h  Sx  =  4x  is  an  identity. 
So,  too,  (x  +  of  =  x'2  +  2ax  +  a?  is  an  identity,  and  is  also  called  a 
formula.     In  such  expressions  x  may  have  any  value. 


SIMPLE   EQUATIONS.  47 

55.  If  x  +  4  =  9,  x  can  have  but  one  value,  5. 

Definition. — A  simple  equation  with  one  unknown 
letter  consists  of  two  expressions  ivith  the  sign  =  between 
them,  in  which  the  unknown  letter  has  a  determinate 
value. 

5ti.  To  solve  an  equation  is  to  find  the  value  of  this  un- 
known letter. 

57.  The  equation  is  satisfied,  or  the  solution  is  verified, 
when  this  value  put  for  the  unknown  letter  in  the  equa- 
tion makes  it  an  identity. 

Thus,  x  +  4  =  9,  gives  x  =  5  ;  and  5  put  for  x  in  the  equation 
gives  5  4-4  =  9,  an  identity,  and  the  equation  is  satisfied. 

58.  The  expression  on  the  left-hand  side  of  the  sign  == 
is  called  the^rs^  side.  The  expression  on  the  right-hand 
side  is  called  the  second  side  of  the  equation. 

59.  Axioms  concerning  equations. 

Axiom  1. — Both  sides  of  an  equation  may  be  multiplied 
or  divided  by  the  same  number,  and  the  equality  still  sub- 
sists. 

Axiom  2.  —  The  same  number  may  be  added  to  or  sub- 
tracted from  both  sides  of  an  equation,  and  the  equality 
still  subsists. 

The  equality  still  subsists  when  we  change  the  signs  of  atl  the 

terms  on  both  sides,  as  this  is  simply  multiplying  both  sides  by  —  1. 


48  SIMPLE  EQUATIONS. 

60.  Transposition. — To  transpose  a  term  is  to  change 
it  from  one  side  of  an  equation  to  the  other. 

Rule. —  When  we  transpose  a  term,  ive  must  at  the  same 
time  change  its  sign. 

This  is  the  same  as  adding  the  same  number  to,  or  sub- 
tracting the  same  number  from,  botli  sides. 

For  example,  if  we  subtract  G  from  both  sides  of  the  equation 

x  +  6  =  15, 
we  have  x  +  6  —  6  =  15  —  G, 

or  x  —  15  —  G. 

Thus,  6  has  been  transposed  from  the  first  side  to  the  second,  and 
its  sign  changed  from  -f  to  —  . 

So,  also,  in  Qx  =  2  —  3x.     To  transpose  3x  to  the  first  side  is  the 
same  as  adding  3x  to  both  sides. 

Thus,  Gx  +  3x  =  2  -  3x  +  Sx, 

or  G.c  +  3x  =  2. 

So  ox  is  transposed  and  its  sign  changed. 

SOLUTION     OF     EQUATIONS. 
Equations  without  Fraction*, 

61.  If  the  equation  has  no  fractions,  it  is  solved  by  the 
following 

Rule. — 1.  Transpose  the  unknown  letters  to  the  first  side, 
and  the  numbers  or  known  terms  to  the  second  side. 

2.  Apply  the  rule  of  addition  to  the  terms  collected  on  the 
tivo  sides. 

3.  Then  divide  loth  sides  by  the  coefficient  of  x.      (Ax- 
iom 1.) 


SIMPLE  EQUATIONS.  49 

Example.  Given     12a;  -  27  =  37  -  3x  +  41,  to  find  x. 

Transposing,  12z  +  Sx  =  37  +  27  +  41. 

Adding  collective  terms,     15#  =  105. 
Dividing  by  lo,  x  =  7. 

Examples — 23. 
Find  the  value  of  a;  in  each  of  the  following  equations  : 

1.  x  +'4  =  10 ;  x  +  24  =  20. 

2.  2a?  +  5  +J*  =  a;  +  14  +  2  ;  9x  =  72. 

3.  2z  +  3a;  =  55  ;  16a;  —  2x  —  6x  =  25  +  4a;. 

4.  x  -  12.5  =  13.  G  ;  12a;  =  104. 

5.  1  +  3a;  +  3  +  5x  =  5  +  7a;  +  7  +  9a\ 

Equations  with  Letters  for  the  Known  Numbers. 

62.  We  often  have  equations  in  which  the  known  num- 
bers are  represented  by  the  letters  #,  b,  c,  etc.  (first  letters 
of  the  alphabet),  as  the  unknown  are  represented  by  x,  y,  z. 
etc.,  the  last  letters. 

Ex.  1.  4a;  +  3«  -  25  =  Via  +  x  -  Sb. 
Transposing,  4z  —  x  —  12a  —  3a  -f  25  —  Sb. 

Collecting,  3x  =    da  —  6b. 

Dividing  by  3,  x  =    3a  -  2b. 

Ex.  2.     mx  —  nx  +  c. 

Transposing,  nix  —  nx  =  c. 

Collecting,  (m  —  n)  x  —  c. 

Dividing  by  coefficient  (co-factor)  of  x,  we  have 

_       c 
~  m  —  n* 


50  SIMPLE  EQUATIONS. 

Examples — 24. 

Find  x  in  the  equations  : 

1.  x  +  a  —  b  ;  x  —  a  =  c. 

2.  2x  —  2a  -{-  c  =  b  ;  ax  —  c. 

3.  ax  —  bx  -f  jp  =  m  —  ft. 

4.  3a;  +  J  —  a  =  5x  +  c. 

Equations  with  Terms  in  Brackets, 

63.  When  the  equation  contains  terms  in  brackets,  the 
brackets  must  be  removed,  as  in  Art.  20. 

Ex.  1.   Qx  -  (2x  -  18)  =  22. 
Removing  the  brackets, 
we  have  Qx  -  2x  +  18  =  22. 

Transposing,  Qx  -  2x  =  22  -  18. 

Reducing,  4x  =  4, 

and  x  =  l. 

Examples — 25. 
Solve  the  equations : 

1.  hx  -  (3  +  2x)  =  12  ;   a  -  9  =  5(x  -  5). 

2.  5(a  -  6)  =  -  40  ;   30  -  2x  =  6x-  (24  -  z). 

3.  1  4-  Sx  -  (2x  -  7)  =  10. 

4.  bx  +  3(4a  -  15)  =  50  -  2x. 

Equations  with  Fractions. 

16.  If  the  equation  has  fractions,  we  first  get  rid  of  the 
fractions.     This  is  done  by  the  following 

Rule. — Apply  the  process  for  reducing  all  the  terms  on 


SIMPLE  EQUATIONS.  51 

both  sides  to  the  least  common  denominator,  dropping  the 
common  denominator. 


„      .,     x       bx       3x       . 


The  L.  C.  D.  is  24. 

Hence  we  have  6x  +  20?  —  9x  =  96. 

173  =  9G, 


96  KIA 

and  a;  =  —  =  5H. 


„      _     2#       SB  —  2         .       x  —  3 
Ex.  2.   —  +  — ^—  =  4  -  — ^r- 


Clearing  of  fractions, 

Ax  +  3(3  -  2)  =  24  -  2(a!  -  3), 

or  4a;  +  33   -  6  =  24  —  23    +  6. 

Collecting,  9a;  =  3G, 

and  3  =  4. 

Ex.3.  '-1      to" 


a;  -  2       4a;  -  7 
Clearing  of  fractions,         (a  —  1)  (4flj  —  7)  =  (43  —  5)  (3  —  2). 
Removing  brackets  (Art.  21),  we  have 

Ax1  -  lis  +  7  =  Ax"  -  13.?  +  10. 
Striking  out  the  common  term  Ax"  from  both  sides,  we  have 

-  llr  +  7  =  -  133  +  10. 
Transposing  and  reducing, 

23  =  3, 

3 

and  3  =  g  . 


52  SIMPLE  EQUATIONS. 

Examples — 26. 
Find  x  in  the  following  equations  : 


2a;       4a;       99  .    x         x  _  2„ 


2.    ^  +  T  =  ^ 
4         5 


3.    |.  +  |+|  =  156. 


*•     a        9   +  6 


„     2x       Zx       ix       „, 
5'    "3    +    5    "  15  =  ** 


a;         a;    _  3a       2x        a_ 
6*    a   +    3~  ~~  ~9   ~  T        5+5 


x  +  12       a;  -  10  =  1_ 

7*         7  10       "     2  ' 

n 2a?       a?  -  2  a?  -  3 

8-  y  +  -2-  =  4-~3-- 

■  3a;  +  4       7a;  -  3  _  x  -  16 

9'  —5 2~~  -— i 

io.  K*  +  6)  -  Tv(ifi  -  to)  =  H. 

ii  A_±=±-l 

"'  a;        4a:       5a;       20 


PROBLEMS  IN  SIMPLE  EQUATIONS.  53 

21 
42—5' 


12. 

6                5 
x  -  2  ~  x  -  3  ' 

12 

2x  +3 

13. 

2x  -3       %x  -  4 
3#  —  4  ~  3z  —  5 ' 

14. 

6a:  -  4        a?  -  2 

21        '    5^-0 

2a; 

7  " 

SECTION     XIX. 

TRANSLATION   OF   ORDINARY  LANGUAGE 
INTO  ALGEBRAIC  EXPRESSIONS. 

65.  As  an  introduction  to  the  solution  of  prob- 
lems by  algebra,  we  give  some  examples  of  translations 
from  ordinary  language  into  algebraic  expressions. 

Ex.  1.  A  man  has  x  dollars,  and  gains  two  dollars. 
How  much  has  he  altogether  ?  Ans.  x  +  2  dollars. 

Ex.  2.  A  man  has  x  dollars,  and  loses  20.  How  much 
has  he  left?  Ans.  x  -  20. 

Ex.  3.  A  man  has  x  dollars,  what  is  one-fourth  of  it  ? 

one-third  of  it  ?  etc.  A  x       x 

Ans.    —r,  -=-,  etc. 

Ex.  4.  If  x  is  the  price  of  a  dozen  oranges,  what  is  the 

price  of  nine  ?  .        9x 

Ans.  j2  . 

Ex.  5.  A  number  exceeds  x  by  7.     What  is  it  ? 

Ans.  x  +  7. 


54  PROBLEMS  IN  SIMPLE  EQ UA  TIONS. 

Ex.  6.  x  exceeds  a  number  by  7.  What  is  that  number  ? 

Ans.  x  —  7. 

Ex.  7.  A  man  has  x  dollars,  and  loses  3  of  them,  and 
then  loses  ^  of  what  is  left.  How  much  has  he  after  both 
losses  ?  Ans.  f  (x  -  3). 

Ex.  8.  x  dollars  at  G  per  cent,  interest  will  yield  how 

much  in  one  year  ?  .  Gx 

Ans.-m. 

(jX 

Will  amount  to  how  much  ?  Ans.  x  +    ^ . 

Ex.  9.   A  man  having  x  dollars  gave  away  ^  of  it,  \  of 

it,  and  ^  of  it ;  how  much  had  he  left  ? 

fx       x        x\  VLx     .       x 

Ans.   a_  (j  +  w  +  ¥z)     or    x-—,t.e.B. 

Ex.  10.  500  is  divided  into  two  parts,  one  of  which  is  x, 
what  is  the  other  ?  Ans.  500  —  x, 

Ex.  11.  If  a  man  goes  x  miles  in  8  hours,  how  many 

miles  per  hour  does  he  travel  ?  .        x 

1  Ans.  3-  . 

o 

Ex.  12.  If  a  man  goes  15  miles  in  x  hours,  how  many 

miles  per  hour  ?  .15 

1  Ans.  — . 

x 

Ex.  13.  If  a  man  buy  x  yards  of  cloth  for  810,  what  is 
the  price  per  yard  ?  .10 

X 

Ex.  14.  If  a  man  pay  x  dollars  a  yard  for  815  worth  of 

cloth,  what  is  the  number  of  yards  bought  ?  15 

Jlns.  - — - . 


PROBLEMS  IN  SIMPLE  EQUATIONS.  55 

Ex.  15.  A  man  goes  x  miles  at  the  rate  of  5  miles  an 

hour ;  what  is  the  number  of  hours  ?  x 

Ans.  =-. 
o 

Ex.  1G.  If  a  man  does  a  piece  of  work  (working  uni- 
formly) in  x  hours,  how  much  of  it  does  he  in  1  hour,  in 
2  hours,  in  3  hours,  in  6  hours,  etc.  ? 

12      3       6 
Ans.  —  ,    —  ,   —  ,    — ,  etc. 

JD  £  'Jb  JO 

Ex.  17.   If  a  man  does  a  piece  of  work  in  10  days,  how 

much  of  it  does  he  in  x  days  ?  .         x 

Ans.  -. 

Ex.  18.  A  pipe  fills  a  cistern  in  8  hours,  what  fraction 
of  it  does  it  fill  in  x  hours  ? 

A  x 

Ans.  — . 
8 

Ex.  19.  What  number  bears  to  x  the  proportion  of  3  to  4  ? 

Ans.  — . 
4 

Ex.  20.  If  two  numbers  bear  to  each  other  the  ratio  of 
4  to  5,  and  one  of  them  is  4%,  what  is  the  other  ? 

Ans.  5x. 

Note.— Consecutive  numbers  are  numbers  each  of  which  is  greater  by  unity 
than  the  preceding  one  ;  thus,  2,  3,  4  are  consecutive  numbers. 

Ex.  21.  Write  four  consecutive  numbers  of  which  x  is  the 
smallest.  Ans.  x,  x  -f  1,  x  +  2,  x  +  3. 

Ex.  22.  Write  three  consecutive  numbers  of  which  x  is 
the  greatest.  Ans.  x  —  2,  x  —  1,  x. 

Ex.  23.  If  £  is  the  tens  figure,  and  y  the  units  figure 
of  a  number,  what  is  the  number  ?  Ans.  lOx  -f  y. 


56  PROBLEMS  IN  SIMPLE  EQUATIONS. 

Ex.  24.  If  x  be  the  number  of  minute  spaces  moved  over 
by  the  minute  hand  of  a  watch  in  a  certain  time,  what 
number  of  spaces  does  the  hour  hand  move  over  at  the 
same  time  ?  ^m     x_  ^ 

Ex.  25.  A  man  travels  25  miles  in  6  hours,  how  many 

miles  does  he  travel  in  x  hours  at  the  same  rate  ? 

.        25# 

Am.   — — . 
o 


SECTION    XX. 
PROBLEMS  IN  SIMPLE  EQUATIONS. 

66.  To  form  an  equation  we  follow  this 

Rule. — Represent  the  unknown  quantity   by   x  ; 
form  the  expressions   and   the  equation  according  to  the 
conditions  of  the  problem. 

The  equation  thus  formed   is   solved  as  explained  in 
Section  XVIII. 

Ex.  1.  What  number  is  that  to  which  if  5  be  added  J  of 
the  sum  will  be  25  ? 

Let  x  =  the  number. 

Adding  5  to  it,  we  have  x  +  5,  and  J  of  this  is  $(x  +  5). 

By  the  condition,  {(x  +  5)  =  25. 

Hence,  x  +  5   =  75, 

and  x   —  70. 

Verification.  £(70  +  5)  =  25, 

or  25  =  25,  an  identity. 


PROBLEMS  IN  SIMPLE  EQUATIONS.  57 

Ex.  2.  What  number  is  it  of  which  the  third  and  fourth 
parts  together  make  21  ? 


Let 

X 

=  the 

number.  . 

Then 

X 

3 

=  its  third  part, 

and 

X 

4 

=    its  fourth  part. 

Then  by  the  questior 

X 

1    3 

X 

+  r 

=  81. 

Reducing, 

4x 

+  3a;  : 

=  21  x  12, 

or 

7X: 

=  252, 

and 

X  : 

=  36. 

Ex.  3.  Find  two  consecutive  numbers  such  that  J  of  the 
smaller  added  to  ^  of  the  greater  is  equal  to  5. 

Let  x  =  the  smaller  ; 

then  x  +  1  =  the  greater. 

x      x  +  1 

Hence  we  have    -r  -\ 5—  =  5. 

4  3 

Reducing,  7x  =  56, 

and  x  =  8. 

Hence,  a;  +  1  =  9,  and  the  two  numbers  are  8  and  9. 

Ex.  4.  Divide  54  into  two  parts,  one  of  which  shall  be 
to  the  other  as  4  :  5. 


Let 

4x  =  one  part, 

and 

5x  —  the  other  part. 

Then 

4ic  +  5x  =  54. 

.'.  x  =  6  ;  4x  =  24,  one  part ;  and  5x  =  30,  the  other  part. 
3* 


58  PROBLEMS  IN  SIMPLE  EQUATIONS. 

Verification.  30  +  24  =  54. 

?i  _1 
30  "  5"' 


Ex.  5.  Divide  eight  dollars  and  a  half  into  the  same 
number  of  dimes,  half-dollars,  and  quarters. 

Let  x  —  the  number  of  each. 

Then  10^  =  the  value  of  the  dimes  in  cents, 

50.r  —  "of  the  half-dollars  in  cents, 

25x  —  "of  the  quarters  in  cents. 

Therefore,  lOx  +  50x  +■  25a;  =  850  ; 

whence  X  =  10. 

Hence,  10  dimes,  10  half-dollars,  and  10  quarters  make  eight  dol- 
lars and  a  half. 

Examples— 27. 

1.  What  number  is  that  which  multiplied  by  7  is  greater 
by  12  than  51  ? 

2.  What  number  is  it,  ^  of  which  is  3  greater  than 
15  ? 

3.  A  train  has  15  more  freight  cars  than  passenger  curs, 
and  33  cars  in  all.     How  many  of  each  sort  in  it  ? 

4.  A  garrison  of  3,280  men  has  3  times  as  many  artil- 
lerists as  cavalry  men,  and  4  times  as  many  infantry  as 
artillerists.     How  many  of  each  of  these  men  ? 

5.  What  number  is  it  J  of  which  added  to  T1¥  of  it  is 
equal  to  20  ? 


PROBLEMS  IJY  SIMPLE  EQUATIONS.  59 

6.  The  difference  between  J  and  J  of  a  number  added 
to  |  of  it  is  22,  what  is  the  number  ? 

7.  Find  two  consecutive  numbers  of  which  \  of  the 
greater  subtracted  from  J-  of  the  lesser  leaves  1. 

8.  Divide  45  into  three  parts  which  shall  be  consecu- 
tive numbers. 

9.  Find  two  consecutive  numbers  of  which  the  lesser 
diminished  by  8  is  one-half  the  greater. 

10.  The  sum  of  two  numbers  is  24,  and  9  times  the  one 
is  equal  to  3  times  the  other.     Find  the  numbers. 

11.  The  difference  of  two  numbers  is  12,  and  4  added 
to  twice  the  smaller  gives  the  greater.  Find  the  two  num- 
bers. 

12.  I  paid  £34  in  half-dollars,  quarters,  and  dimes,  and 
used  the  same  number  of  each  of  these  coins.  What  was 
this  number  ? 

13.  A  father  is  now  4  times  as  old  as  his  son.  Five 
years  ago  lie  was  5  times  as  old;  what  is  now  the  age  of 
each  ? 

14.  A  company  consists  of  90  persons  ;  the  men  are  4 
more  than  the  women,  and  the  children  10  more  than 
the  grown  persons.     Find  the  number  of  each. 

15.  The  \,  \,  and  |  of  a  certain  sum  of  money  are  to- 
gether £4  more  than  the  amount  itself.     What  is  it  ? 


00  PROBLEMS— CONTINUED. 

SECTION    XXI. 
PROBLEMS— Continued. 

67.  Ex.  1.  A  can  do  a  piece  of  work  in  4  days,  and  B 
can  do  it  in  3  days.     In  what  number  of  days  will  they 

both  together  do  it  ? 

Let  the  work  be  1,  and  x  =  the  required  number  of  days. 

x 
In  x  days  A  does  j  of  the  work. 

x 
In  x  days  B  does  ~-  of  the  work. 
o 

Hence,  _  +       —  1 

4        6 

or  7x  ■—  12, 

and  x  =  ly  days. 

Ex.  2.  A,  B,  and  C  divide  700  acres ;  A  taking  4  acres 
to  B's  5,  and  3  acres  to  C's  2.  How  many  acres  did  each 
get? 

Let    x  —  A's  number  of  acres; 
then      \x  -  B's 
and       \x  —  C's        "  " 

Hence,  x  +  f  x  +  \x  —  700. 

Solving,  x  —  240  =  A's  acres. 

\x  =  300  =  B's     " 
lx  =  160  =  C's     " 

Ex.  3.  A  man  had  £2,000,  a  part  of  which  he  lent  at  4 
per  cent,  per  annum,  and  a  part  at  6  per  cent.  The  an- 
nual income  from  the  whole  was  $92.     Find  the  two  parts. 


PROBLEMS— CONTINUED.  61 

Let  x  =  the  number  of  dollars  lent  at  4  per  cent.     This  produces 

Ax 
zr^.  dollars  per  annum. 


2000  —  x  =  the  number  of  dollars  lent  at  6  per  cent,  and  this 
6_ 
100 


yields  — —  (2000  —  x)  dollars  per  annum. 


4r  fi 

Therefore,  m  +  m  (2000  -  x)  =  92 

whence  x  =  1400  dollars, 

and  2000  -  x  =  600  dollars. 


Examples — 28. 

1.  Divide  102  into  4  parts  which  shall  be  consecutive 
numbers. 

2.  A  cistern  is  filled  by  one  pipe  in  8  hours,  and  by  an- 
other in  3  hours — in  what  time  will  it  be  filled  if  both 
pipes  run  at  the  same  time  ? 

3.  Find  a  number  such  that  if  the  half  of  it  be  taken 
from  3G,  ^  of  the  remainder  will  be  equal  to  f  of  the 
original  number. 

4.  A  and  B  being  on  the  same  road  21  miles  apart, 
they  set  out  at  the  same  hour  towards  each  other,  A  walk- 
ing 3  miles  an  hour  and  B  at  the  rate  of  4  miles  an  hour. 
How  many  hours  will  elapse  ere  they  meet,  and  how  far 
will  each  have  walked  ? 

5.  A  can  do  a  piece  of  work  in  5  days,  B  in  6  days,  and 
C  in  8  days.  In  what  time  can  they  do  it  all  working  to- 
gether ? 


62  PR0BLE3IS-C0JSTINUED. 

6.  A  farmer  had  two  flocks  of  sheep  of  the  same  num- 
ber. He  sold  39  from  one  flock,  and  93  from  the  other, 
and  found  he  had  remaining  in  one  flock  twice  as  many  as 
in  the  other.  How  many  were  in  the  flocks  at  the  be- 
ginning ? 

7.  A  sets  out  for  a  town  12  miles  off,  and  walks  at  the 
rate  of  4  miles  an  hour.  Half  an  hour  afterward  B  sets 
out  from  the  same  place,  in  the  same  direction,  running  5 
miles  an  hour.   How  far  from  the  town  will  B  overtake  A? 

8.  The  denominator  of  a  certain  fraction  exceeds  the 
numerator  by  2,  and  if  2  be  subtracted  from  the  numerator 
and  added  to  the  denominator,  the  new  fraction  thus 
formed  is  equal  to  J.     What  is  the  fraction  ? 

9.  Seven  maidens  met  a  boy  who  was  carrying  a  basket 
of  apples.  One  maiden  bought  f  of  the  apples ;  the 
second,  -fa  ;  the  third,  -J- ;  and  the  fourth,  ^  of  them  ; 
the  fifth  bought  20  apples  ;  the  sixth  bought  12,  and  the 
seventh  bought  11,  and  this  left  the  boy  one  apple.  How 
many  had  he  at  first,  and  how  many  did  the  first  four 
maidens  take  ? 

10.  Polycrates,  the  tyrant  of  Samos,  asked  Pythagoras 
the  number  of  his  pupils.  Pythagoras  answered  him  : 
The  half  of  them  study  mathematics  ;  one-fourth  part 
study  the  secrets  of  nature  ;  the  seventh  part  listen  to  me 
in  silent  meditation,  and  then  there  are  three  more,  of 
whom  Theano  excels  them  all.  This  will  give  you  the 
number  of  pupils  whom  I  am  guiding  to  the  boundaries  of 
immortal  truth. 


TWO   UNKNOWN  QUANTITIES— ELIMINATION.      63 


SECTION     XXII. 

SIMPLE  EQUATIONS   WITH  TWO   UNKNOWN 
Q  UANTITIES.—ELI3IINA  TION. 

68.  Simultaneous  Equations  are  those  winch  are 
true  for  the  same  values  of  the  unknown  quantities  which 
they  contain. 

If  x  —  y  =  5,     or    x  =  y  +  5  .     .     .     .    (1). 

Then,  when  y  =  1,  x  =  G  ; 

y  =  2,  x  =  7  ; 

y  ■—  3,  x  =  8,  etc.,  indefinitely. 

Now  suppose,  at  the  same  time, 

»  +  y  = » (^). 

Then  among  the  values  of  #  and  ?/  in  the  above  list, 
only  x  =  2,  y  =  7  will  satisfy  equation  (2),  and,  therefore, 

I  are  the  only  values  which  belong  to  both  of  the 

x  —  y  —  5  ) 
equations  [■  taken  together,  or  simultaneously. 

x  +  y  =  9  )  J 

Hence,  2V;o  simultaneous  simple  equations  with  two 
unknowns  give  one  value  for  each  of  the  two  unknowns. 

69.  Elimination.— To  find  these  values,  we  first  re- 
duce the  two  equations  to  a  single  (qua Hon  containing 
only  one  of  the  unknown  letters.  This  process  is  called 
elimination.  That  ie,  we  first  eliminate,  or  get  rid  of  one 
of  the  unknown  letters.  Three  methods  of  elimination 
are  usually  given. 

70.  First  Method. — Multiply  the  given  equations  by 


64       TWO  UNKNOWN  QUANTITIES— ELIMINATION. 

such  numbers  as  will  make  the  coefficients  of  one  of  the  un- 
knowns the  same  in  both.  Then  add  or  subtract  the  equa- 
tion thus  obtained,  according  as  these  equal  coefficients 
have  contrary  or  the  same  signs.  We  thus  get  a  simple 
equation  with  one  unknown  letter. 

Ex.  1.   Given  %x  +  3y  =  9     (1) ) 

,  ,  r  to  find  x  and  ?/. 
4z  -  by  =  7     (2)  )  J 

To  eliminate  y,  multiply  equation  (1)  by  5,  and  equation  (2)  by  3. 

We  have  this. 

lOz  +  15y  =  45 

12:r  -  15y  =  21 

Adding,  we  get  22#  =  66 

.-.     x  -  3. 
We  may  find  y  by  eliminating  x,  or  more  simply  thus  : 
Since  x  ■=  3,  2x  —  6  ;  and  hence,  substituting  6  for  2x  in  (1),  we 

get 

6  +  Sy  =  9, 

or  y  =  1. 

Therefore,  *  =  3   i  . 

are  the  required  values  of  x  and  y. 
y  =  1   ) 

Ex.  2.   Given  2x  +  3w  =  13     (1)' )  x    a    , 

4z  +  2t/  =  14     (2)  5  9 

To  eliminate  x,  we  have 

4,r  +  Qy  =  26 

ix  +  2y  =  14 

Subtracting,  we  get  Ay  =  12 

or  y  =  3. 

And  putting  3  for  y  in  (1),  we  have 

2*  +  9  =  13, 
or  x  —  2. 


TWO  UNKNOWN  QUANTITIES— ELIMINATION.       65 

71.  Second  Method.  —Find  the  expression  for  one  of 
the  unknown  letters,  as  x,  in  terms  of  the  other,  in  one  of  the 
equations,  and  substitute  this  for  x  in  the  other  equation. 

Ex.  3.     Sx  -    y  =    6 (1). 

5x  +  %y  =  32 (2). 

The  expression  for  x  in  (1)  is  x  =  *—= —  . 

o 

Putting  this  for  x  in  (2),  we  have 

5  (^f8)  +  2V  =  32, 

or  5y  +  30  +  6y  =  96, 

lly  =  66, 
and  y  =    6. 

Again,  putting  6  for  y  in  (1),  we  have  3x  —  6  =  6, 
or  3a;  =  12     and    x  =  4. 

72.  Third  Method. — Express  one  of  the  unknown 
letters  in  terms  of  the  other  in  each  equation,  and  put  these 
expressions  equal  to  one  another. 

Ex.  4.  Given  4a  +  Sy  =  19   .     .     .     .     (1). 
2<k  -  5y  =    3   .     .     .     .     (2). 

1Q  _  4<r 
From  (1),       Sy  =  19  -  Ax,     or    y  =  — *--    ....     (3). 


(4). 


From  (2),       5y  = 

=  2a; 

- 

3,       or 

y  = 

2x 

-3 
5 

• 

• 

Equating  (3)  and 

(4), 

19 

—  Ax 
~3        ~ 

2x  - 
~ 5 

■  3 

» 

01 

Hence, 

95- 

-20x  = 
26a?  = 

X  — 

6a;- 
:  104, 
4. 

9. 

Putting  4  for  x  in  (2), 

we 

get  8  - 

5«  = 

0; 

hence, 

y  = 

=  1. 

66       TWO  UNKNOWN  QUANTITIES— ELIMINATION. 

Examples— 29. 
Solve  the  simultaneous  equations 


x  +    y  —  15  ) 
Ox    =  4y)  ^' 


x-    y  =    30 

2x  +  y  =  120 


f(3). 


bx  +  9y  =  65  j 
7a;  +  3y  =  43  f  (5)* 


7a;  +  2#  =  85 
18a  —  3y  =  129 


|  (7). 


11 


+  2y 


15  1 


5a;  +  ^  =  58 


K9). 


X  + 
X  — 

y  = 
y  = 

=  24) 

=  io  f (3>- 

X  + 

3x- 

4?/  = 

=  14) 

=4  m 

8a;  + 
12a;- 

3y, 
9y  = 

=  113) 

X 

4  + 

a; 

2  + 

3  " 

4  " 

K8). 

=7\ 

16a;  + 
24a* -1 

17y  = 
05y  = 

=  274/ 

7*-32,  =  0  ,■ 

2x  +  5y  =  41  f  l  ;* 


5(z+2)-3(y  +  l)=23| 

3(a;-2)+5(7/-l)  =  19i(12)- 


3  +  10  ~  G7° 


3a; 


W13). 


-  y  =  1250 


3x  =  2y  +  14 
J5_  _  200 
x  —  y~"    x 


PROBLEMS— SIMULTANEOUS  EQUATIONS.         67 

—3-+    8y=    31  1  1.7*-a.2y=-7.9) 

HIS).  ,  K        1  M16)- 

y—-^  +  10*;  =  192  ^ 

4  J 


a  -  1       y  +  2  _  2(x  -  y)  }  x  ) 

"~3 T~ 5~"  3~7J=Z    °l 

H^)-  His). 

?^5  _  lSl^  -  2v  -x  x  -  t  -  10 


SECTION    XXIII. 

PROBLEMS  PRODUCING  SIMULTANEOUS 
EQUATIONS. 

73.  Ex.  1.  The  sum  of  two  numbers  is  21,  and  their 
difference  is  5.     Find  the  numbers. 


Let 

x  —  one  number, 

and 

y  —  the  other  number. 

Then 

X  +  y  =  21 1 

x  —  y  —  5   ) 

Adding, 

2z  =  26, 

and 

B=  13. 

Putting  13  for  s»,  13  +  y  =  21, 

From  which  we  find  y  =    8, 


68         PROBLEMS—SIMULTANEOUS  EQUATIONS. 

Ex.  2.  The  sum  of  two  numbers  is  44,  and  if  \  of  the 
less  be  added  to  J  of  the  greater,  the  result  is  12.  Find 
the  numbers. 

Let  x  =  the  greater  number, 

and  y  =  the  less  number. 

Then  x  +    y  —  44 

\x  +  \y  =  12 

From  which  we  find  #  =  32, 

y=  12. 

Ex.  3.  A  certain  number  consists  of  two  digits.  The 
sum  of  the  digits  is  7,  and  if  45  be  added  to  the  number, 
we  get  a  number  the  digits  of  which  are  the  digits  of  the 
first  number  reversed.     Find  the  number. 


Let 

x  =  the  tens  digit, 

and 

y  —  the  units  digit. 

Then 

10#  +  y  =  the  number, 

and  we  have 

x  +  y  =  7             ) 

10a 

+  y  +  45  =  lOy  +  x  ) 

which  give 

X 

=  1,       y=    6; 

or  the  number  is  16  ;  and  1G  +  45  —  61,  the  digits  of  16  reversed. 


Examples — 30. 

1.  Find  two  numbers  such  that  -J  of  the  one  added  to  £ 
of  the  other  shall  be  20,  but  ^  of  the  latter  added  to  J  of 
the  former  shall  be  equal  to  22. 


PROBLEMS— SI3WLTANE0US  EQUATIONS.         69 

2.  On  adding  18  to  a  certain  number  of  two  digits,  we 
get  a  number  in  which  these  digits  are  reversed,  and  the 
two  numbers  added  together  make  44.  What  is  the  first 
number  ? 

3.  Divide  46  into  two  such  parts  that  when  the  greater 
part  is  divided  by  7,  and  the  smaller  part  by  3,  the  quo- 
tients together  make  10. 

4.  A  said  to  B,  give  me  f  of  your  money  and  I  will  have 
$100.  B  said  tc  A,  give  me  J  of  yours  and  I  will  have 
8100.     Find  how  much  A  and  B  had  at  first. 

5.  The  difference  between  two  sums  put  out  at  interest 
for  one  year  is  $2,000.  One  sum  is  put  out  at  5  per  cent, 
and  the  other  at  4  per  cent. ,  and  the  incomes  from  them 
are  equal.     Find  the  two  sums. 

6.  A  merchant  has  his  house  and  goods  together  insured 
for  $36,000.  His  house  at  1|  per  cent,  premium,  and  his 
goods  at  2  per  cent.  The  difference  of  the  two  premiums 
amounts  to  $6.75.  What  was  the  amount  of  insurance  .on 
the  house,  and  what  on  the  goods  ? 

7.  What  fraction  is  that  which  is  equal  to  TV  when  2  is 
subtracted  from  its  numerator,  and  equal  to  -J-  when  4  is 
added  to  its  denominator  ? 

8.  If  the  number  of  cows  in  a  field  were  doubled,  then 
there  would  be  84  cows  and  horses  together  in  the  field. 
If  the  number  of  horses  be  doubled,  and  f  of  the  cows  be 
taken  out  there  would  be  80  horses  and  cows  in  the  field. 
What  is  the  number  of  horses  and  cows  in  the  field  ? 


70    EQUATIONS  OF  THREE  UNKNOWN  QUANTITIES. 

9.  The  sum  of  two  numbers  is  30,  and  their  quotient 
is  4.     Find  the  numbers. 

10.  In  the  second  class  of  a  school  there  are  1-J-J  times 
as  many  pupils  as  there  are  in  the  first  class.  From  the 
first  class  7  go  away,  and  6  enter  the  second  class,  and  then 
the  second  has  2f  as  many  as  the  first.  How  many  in 
each  class  ? 


SECTION     XXIV. 

SIMULTANEOUS     SIMPLE     EQUATIONS     OF 
THHEE  OB  MOliE    UNKNOWN  QUANTITIES. 

74.  If  three  equations  are  given,  with  three  unknown 
quantities,  we  may  eliminate  one  of  the  unknown  quan- 
tities, and  thus  obtain  two  equations  with  two  unknowns, 
and  then  from  these  two  we  may  obtain  one  equation 
with  one  unknown,  by  the  methods  of  Arts.  70,  71,  72. 

Example.  Given 

7x  +  2tj  +  Zz  =  20  (1)  ^1 

3x  —  fy  +  2z  =    1  (2)  )■  ,  to  find  %,  y,  and  z. 

-2x  +  5y  +  7z  =  29  (3)  J 

First,  to  eliminate  y  between  (1)  and  (2), 
we  have  14a;  +  4?/  +  6z  =  40 

3x  —  4//  +  2z  =    1 


Hence,  17a;  +  Sz  =  41    (4) 


EQUATIONS  OF  THREE  UNKNOWN  QUANTITIES.     H 

Second,  to  eliminate  y  between  (1)  and  (3), 
we  have  35a  +  1%  +  I02  =  100 

-  4a  +  lOy  +  14s  =    58 


Hence,  39a  +  z  =    42  (5). 

Third,  to  eliminate  z  between  (4)  and  (5), 
we  have  312a  +  Sz  =  336 

17a  +  8z  =    41 


Hence,  295a  =  295, 

or  a  =  1. 

Then,  from  (4)  17  +  Sz  =  41, 

or  Sz  =  24, 

and  z  =  3. 

And  from  (2),        3  —  4y  +  6  =  1, 

or  4y  =  8, 

and  y  =  2. 

Examples — 31. 
Solve  the  simultaneous  equations 

bx  4-  4y  -  2z  =  14  ^ 

3a:  +  2y  +    z  =  16  J.  (1). 

I 
a;  -  9//  +  82  =    7  J 

#  +    y  +    2?  =    0  1 

#  —    #  +    2=    4  J>  (2). 
5#  +    y  +    z  =  20 


72    EQUATIONS  OF  THREE  UNKNOWN  QUANTITIES, 

x  +  y  =    8} 


y  +  z  =    6 

K3). 

x  +  z  =  10  J 

y-X  =  iz 

z  -  x  —  \y 

►(*)■ 

x  +  %  =  2(y  -  1) , 

2+3=9 

,(5). 

z       x        r 
5+2=5> 

3a  +  4y  +    z  --  14  "| 

2a;  +    y  +  52  =  1!) 

►(6) 

5z  +  %y  +  3z 

=  18, 

40a:  -  1G#  +  25s  =    6  1 
12a;  +    4ij  +  20z  =  02  V  (7). 
36a;  +  17?/  —  152;  =  5G  ^ 


INVOLUTION  OR  RAISING    TO  POWERS.  73 

SECTION    XXV. 

INVOLUTION  OB  RAISING  TO  BOWERS. 

75.  Involution  is  the  process  of  raising  quantities  to 
powers.  It  is  multiplication,  therefore,  in  which  the  fac- 
tors are  all  equal,  and  requires  no  rules  different  from 
those  already  given  (Section  IV.). 

76.  We  will  notice  some  results,  however,  which  will  be 
of  use  in  shortening  the  process. 

(1.)  Rule  of  the  Signs. — Any  evenpoiver  of  a  minus 
quantity  is  plus.  And  any  odd  power  of  a  minus  quan- 
tity is  minus. 

Thus,  the  square  of  —  a  is  —  a  x  —  a  =  -f  a~. 

The  fourth  power  of  —  a  is  —  a  x  —  a  x  —  a  x  —  a 
=  +  a\ 

The  third  power  of  —  a  is  —  a  x  —  a  x  —  a  =  —  a\ 

The  fifth  power  of  —  a  is  —ax  —ax  —ax  —ax 

—  a  =  —  a\ 

(2.)  Rule  of  the  Exponents. — To  raise  a  quantity 
to  a  power,  multiply  its  exponent  by  the  exponent  of  the  re- 
quired power. 

Thus,  (a")2  —a'  x  a*  =  a*  =  a*** ; 
(a3)2  =  a3  x  a5  —  a6  =  rr*2 ; 
(a*)*  =  a1  x  a*  =  a*  —  a**\ 


74  INVOLUTION  OB  RAISING  TO  POWERS. 

77.  Powers  of  Monomials. — To  raise  a  monomial 
to  a  power, 

Rule. — Raise  the  coefficient  to  the  required  power,  ob- 
serving the  rule  of  the  signs,  and  multiply  the  exvonent  of 
each  letter  by  the  exponent  of  the  power. 

Thus,  the  square  of  a~bz  is  a4b\ 

The  third  power  of  -  2ab  is  -  SaW. 

The  fourth  power  of  2ab°~c2  is  16«4&V2. 

78.  Squares  of  Monomials. —  As  we  will  have 
mainly  to  do  with  squares  in  this  book,  we  will  particular- 
ize for  this  case.     Therefore,  to  square  a  monomial, 

Rule. — Square  the  coefficient  and  multiply  the  exponents 
by  two. 

Thus,  the  square  of  5a*b*c  is  25#fi#V2. 

79.  Powers  of  Fractions. — To  raise  a  fraction  to  a 
power 

Rule. — Raise  the  numerator  and  denominator  to  the  re- 
quired  power,  observing  the  rule  of  the  signs. 


.  2  V       2         2         4 
Thus,    (_  )=¥xT=y 


a  \  _  a         a'  _    a 

TV  ~  T  x  T3  ~  T 


HI) 


INVOLUTION  OR  RAISING  TO  POWERS.  75 

80.  Squares  of  Fractions. — As  a  particular  case,  to 
square  a  fraction, 

Rule. — Square  the  numerator  and  denominator. 

81.  Powers  of  Polynomials. — To  raise  a  polyno- 
mial to  any  power,  we  simply  perform  the  multiplications 
indicated  by  the  exponent  of  the  required  power  (always 
one  less  than  the  exponent). 

Thus,  (a  +  by  =  (a  +  b)  (a  +  b). 

(a  +  b)z  =  (a  +  b)  (a  +  b)  {a  +  b). 
(a  +  by  =  (a  +  by  x  (a  +  6)2  = 
(a  +  J)  (a  +  6)  (a  +  b)  (a  +  b). 

82.  Squares  of  Polynomials. — For  the  squares  of 
polynomials,  we  repeat  the  two  important  results  of 
Articles  25,  26. 

(a  +  b)2  =  a2  +  2ab  +  b*  =  (-  a  -  b)\ 

(a  -  by  =  a'  -  2ab  +  b*  =  (b  -  «)2. 

Also,  by  performing  the  multiplications  required,  we 
find  the  following  useful  results, 

(a  +  b  +  cy  =  a"  +  2ab  +  2ac  +  b-  +  25c  +  c2. 
(«  -  b  +  c)2  =  a"  -  2ab  4-  2«c  +  b*  -  2fo  +  c2. 

Examples— 32. 

1.  Find  the  third  power  of  3ab\ 

2.  Find  the  third  power  of  —  2a2bc. 


76  EVOLUTION  OR  EXTRACTION  OF  ROOTS. 


3.  Find  the  fourth  power  of  ^  . 


Square  each  of  the  following  quantities  : 

3  ex 

2ay  '  2x6 


4.    -Sab.        5.  8crbc\        6.  |--.         7.  -  ^ 


Write  the  squares  of 
8.  a  +  2.         9.  2bc  +  1.         10.  3m  —  5«. 

11.   #C£  +  #•  12.   —  +  c. 

o 

13.  Find  the  square  of  2a  +  3b  —  c. 

14.  Find  the  third  power  of  a  —  £. 

15.  Find  the  fourth  power  of  a  +  b. 


SECTION   XXVI. 


EVOLUTION  OB  EXTRACTION  OF  ROOTS.— 
SQUARE  ROOT. 

83.  Roots  of  Quantities. — The  root  of  a  quantity  is 
the  quantity  the  involution  of  which  produces  the  given 
quantity. 

Thus,  the  square  root  of  a*  is  a,  because  a  squared 
gives  a2 ;  also,  the  cube  root  of  8a3  is  2a,  because  (2a)3  = 
2a  x  2a  x  2a  =  $a\ 

The  fourth  root  of  16a*  is  2a,  because  (2a)*  =  2a  x 
2a  x  2a  x  2a  =  16a4. 


EVOLUTION  OR  EXTRACTION  OF  ROOTS.  77 

84.  Evolution  is  the  process  of  finding  the  roots  of 
quantities,  and  is  the  reverse  operation  of  involution ; 
and  the  rule  of  evolution,  or  extraction  of  roots,  is  found 
from  the  results  of  raising  to  powers. 

85.  The  signs  of  evolution,  or  of  roots  to  be  extracted, 
are  placed  on  the  left  of  the  quantities,  and  are  as  follows  : 

V  for  "  the  square  root  of," 

y'-      for  "  the  cube  root  of," 

J/         for  "fourth  root  of,"  etc.,  etc. 

3,  4,  etc.,  being  called  indices  of  the  roots  (the  index  2 
being  understood  when  none  is  written). 

86.  We  will  notice  some  results  which  follow  directly 
from  the  rules  of  involution. 

(1.)  Any  even  root  of  a  +  quantity  may  be  either  +  or 
—,  and  must,  therefore,  be  written  with  the  double  sign  ± 
(plus  or  minus). 

Thus,     \/9  =  ±3,         V4^  =  ±  2a,        ^/aT=  ±  a. 

(2.)  Any  odd  root  of  a  quantity  has  the  same  sign  as 
the  quantity  itself 

Thus,  \/a3  —  +  a, 


and  V—  «3  —  —  a. 

(3. )  There  can  be  no  even  root  of  a  minus  quantity. 
Thus,  the  square  root  of  —  a2  cannot  be  extracted,  for 


78  EVOLUTION  OR  EXTRACTION  OF  ROOTS. 

(+  af  =  +  a%  and  (—  a)'2  =  +  a'  ;  and,  therefore,  such 
expressions  as  V—  d2  ?  are  called  impossible,  or  imaginary 
quantities. 

SQUARE    ROOT. 

87.  Under  evolution  we  shall  only  give  the  rules  for 
finding  the  Square  Root  of  algebraic  quantities. 

88.  Square  Root  of  Monomials. — The  square  of 
2«3Z>2  is  2«3&2  x  2a3b'2  —  Aa6b*  ;  hence  the  square  root  of 
4«6Z>4  is   ±2a*b\     (Art.  86  (1).) 

Therefore,  to  find,  the  square  root  of  a  monomial,  we 
have  the 

Rule. —  Take  the  square  root  of  the  coefficient,  and  di- 
vide the  exponents  of  the  letters  by  2.  Write  the  sign  ± 
before  the  result. 

Note.— A  minus  quantity  has  no  square  root.     (Art.  86  (3).) 

89.  Square    Root   of  Fractions.— The  square  of 

2a   .4a2  ,,  ,      -  4«a   .      ,   2a 

±  —    is  op  ;  hence  the  square  root  oi  ^  is   ±^-,   or, 

to  extract  the  square  root  of  a  fraction, 

Rule.—  Take  the  square  root  of  the  numerator  and  de- 
nominator. 

16a6  x2  4:a*x 

Ex.  1.   The  square  root  of  _  ,,  ,  is    ± 


Ex.  2.  Find  Vf  +  *#■• 

First  add  these  fractions  :   f  +  a3a 


Hence,    V§  +  ¥  =   W  =  ±  I- 


EVOLUTION  OR  EXTRACTION  OF  ROOTS.  79 


Exampl 

es— 33. 

ind  the  square  root  of  the 

following  quantities  : 

1. 

25a2u\ 

2. 

lOOftVj/2. 

3. 

49a?b\ 

4. 

9ftV 

4*y  * 

5. 

Wtfb* 

6. 

$xy 

±9xy  ' 

~Ja*   ' 

7. 

19ft2       3a- 
~16"  +  ~8~* 

8. 

6         2 

25  +  y 

9. 

4         1 
9         3' 

10. 

37        3 
16      7" 

90.  Square  Root  of  Trinomials. — Since  («  +  by— 
a2  h-  2ft#  -f-  &2,  the  square  root  of  a2  +  2ab  -f  b2  is  ±  (ft  -f  &). 

Since  (a  —  b)2  =  ft2  —  2ab  -f  Z>2,  the  square  root  of 
«2  -  2ft£  +  b2  is   ±  (ft  -  J). 

Therefore,  to  find  the  square  root  of  a  trinomial,  we 
have  the 

Rule. — Arrange  it  with  reference  to  one  of  its  letters, 
then  take  the  square  roots  of  the  first  and  last  terms,  and 
put  the  sign  of  the  middle  term  between  them. 

Ex.  1.  The  square  root  of  xx  +  Gx  +  9  =  a/s?  4-  a/9  = 
±  (a?  +  3). 

Ex.  2.  The  square  root  of  4ft2  -  12  ab  +  W  =  Via*  - 
VW  -  ±  (2a  -  3*). 


80         EVOLUTION  OR  EXTRACTION  OF  ROOTS. 

91.  Note  1. — Any  number  or  algebraic  expression  is  called  a  per- 
fect or  complete  square  when  its  square  root  can  be  exactly  found. 

Note  2. — A  trinomial  is  a  complete  square  if,  when  arranged 
by  one  of  its  letters,  the  middle  term  is  twice  the  product  of  the  square 
roots  of  the  first  and  last  terms;  that  is,  when  the  square  of  the  middle 
term  is  four  times  the  product  of  the  first  and  last  terms. 

Note  3. — Since  a  monomial  squared  gives  a  monomial,  and  a  bino- 
mial squared  gives  a  trinomial,  a  binomial  cannot  be  a  complete 
square. 

Examples — 34. 

Find  the  square  root  of  the  following  trinomials  : 

1.  a2  +  %a  +  1. 

2.  of  +  4  —  4=x. 

3.  x2  +  5x  +  -\K 

4.  9a*b2  -  Qabx  4-  x\ 

5.  16a4  -  2±aW  +  W. 

6.  64a;4  +  f  62  -  2Ux\ 

7.  a2  +  i  +  a. 

8.  16V  -8a;  +  1. 

9.  x2  -  3x  +  f, 

10.  Are  x2  -  6x  +  4,  a2b2  -  2abx  +  x\  a2  -  \a  +  f 
complete  squares  ? 

92.  Completing  the  Square.— In  equations  we  have 
often  to  make  such  algebraic  expressions  as  xu  +  px,  and 


EVOLUTION  OR  EXTRACTION  OF  ROOTS.  81 

x2  —  px,  complete  trinomial  squares  by  adding  the  right 
term.     This  process  is  called  Completing  the  Square. 

Examining  the  trinomial  squares  given  in  Art.  90,  yiz., 
a~  +  2ab  +  ¥ ,  a2  —  2ab  +  b2,  we  see  that  in  order  to 
make  any  binomial  expression  of  the  form  x2  +  px,  or 
x2  —  px  a  complete  trinomial  square,  we  have  the 

Rule. — Add  the  square  of  half  the  coefficient  or  factor 
of  x  in  the  second  term. 

Ex.  1.  To  x2  +  4:X  add  2%  and  we  have  x2  +  4#  +  4, 
the  square  root  of  which  is  ±  (x  +  2) . 

Ex.  2.  To  x1  —  Sax  add  (-«-)>  and  we  have  x1  —  3ax  + 

—  ,  the  square  root  of  which  is  ±  Ix  — x- 

Ex.  3.  To  x2  —  %x  add  (f  )2,  and  we  have  x2  —  fa;  +  $, 
the  square  root  of  which  is  ±  (x  —  |). 

E  x  ample  s — 3  5 . 

Complete  the  squares  of  the  following  expressions,  and 
find  the  square  root  of  each  result. 


1. 

x2  +  6x. 

2. 

x2  -  12a. 

3. 

x1  -  lias. 

4. 

a;2  —  a. 

5. 

f  +  3y. 

6. 

a;2  —  px. 

7. 

a2  -  fa. 

8. 

x2  —  4#2. 

9. 

yf  +  4*A 

10. 

a:2  -  }x. 

1. 

/v2    9   /y 

12. 

x2  +  fa?. 

4* 


82         EVOLUTION  OR  EXTRACTION  OF  ROOTS. 

93.  Square  Root  of  Polynomials.— The  rule  for 
the  square  root  of  polynomials  is  similar  to  the  rule  in 
arithmetic. 

We  know  that  the  square  root  of  a2  +  2ab  +  b2  is 
a  +  b. 

Hence,  writing  down  the  terms,  and  proceeding  as  in 
arithmetic,  Ave  have  the  following  arrangement : 

a2  +  2ab  -f  b2\a  +  b 


a2  2a 


2a  +  b  )  2ab  +  b2 
2ab  +  b2 


That  is,  1.  Arrange  the  polynomial  with  reference  to 
one  of  its  letters. 

2.  Find  the  square  root  of  the  first  term,  and  subtract 
its  square  from  the  polynomial. 

3.  For  a  divisor  double  the  first  term  of  the  root.  Di- 
vide the  first  term  of  the  remainder  by  this  divisor  ;  the 
quotient  will  be  the  second  term  of  the  root. 

4.  Place  this  second  term  in  the  root,  and  lo  the  right  of 
the  divisor.  Multiply  the  divisor,  thus  increased,  by  the 
second  term  of  the  root,  and  subtract  the  product  from  the 
dividend. 

91.  If  the  root  contain  more  than  two  terms,  a  like  pro- 
cess continued,  doubling  each  time  the  root  already  found, 
will  find  all  the  terms. 

Thus,  we  know  (a  +  b  +  a)'  =  a2  +  2ab  +  b2  +  2ac  4- 
2lc  +  c\ 


EVOLUTION  OB  EXTRACTION  OF  HOOTS.  8£ 

Hence,  to  find  the  square  root  of  this  latter  exjn'ession, 
we  proceed  as  in  the  following  arrangement : 

a2  +  2ab  -f  ¥  -f  2ao  -f  2bc  -f  c2 1  a  +  b  4-  c 


a2 

2a +  b    |    2ab  +  b'1 

2ab  +  62 

2a  +2b  +  c   | 

2ae  -f  2Z>6*  +  c8 

2ac  +  2bc  +  c- 

2a 


Examples— 36. 
Find  the  square  root  of 

1.  64a2  +  lUab  +  SlbK 

2.  a;4  +  4a#s  +  Qxfx1  +  4a8#  +  a\ 

3.  a4  -  4^s  +  Sx  +  4. 

4.  9a2  -  12afl  4-  4£2  -f  Gac  -  4fo  +  c\ 

5.  9a4  -  12ar'  f  10a2  -  4a  +  1. 

6.  10a4  +  W  4-  &  -  lGcrb  -f  8aV  -  4#c*. 

7.  9c4  —  6ar  +  a-  +  12c2  -  4a  +  4. 

8.  a2  +  Z>J  --  2ab  +  4ac  —  4fo  +  4c2. 

9.  xk  -  2x%  -  x-  +  2a;  +  1. 


io.  f-i^  +  i. 


11.  a4  -  10a3  +  37a2  -  60a  +  36. 


84  QUADRATIC  EQUATIONS. 

SECTION    XXVII. 

QUADRATIC  EQUATIONS. 

£5.  A  Quadratic  Equation  is  an  equation  of  the 
second  degree,  that  is,  it  contains  the  second  power  of  the 
unknown  letter. 

96.  There  are  two  sorts  of  quadratic  equations  : 
1st.  Pure  Quadratics,  which  contain  #2  and  not  tf. 
2d.  Affected  Quadratics,  which  contain  both  #2  and  x. 

Thus,    5z2  -  2  =  6x>  -5,     x"  =  9,     _  +  —  =  1    are 

pure  quadratics  ; 

x1 
While  3#2  +  5x  —  6, 9x  =  5,     ax*1  +  Ix  =  c    are 

affected  quadratics. 

97.  Solution  of  Pure  Quadratics.— To  solve  a 
Pure  Quadratic, 

Rule. — Find  x2,  as  in  simple  equations.  Then  tafce 
the  square  root  of  both  sides, putting  the  sign  ±  before  the 
root  of  the  second  side. 

Ex.  1.  Given   x>  -  ~  =  1. 
4 

Clearing  of  fractions, 

4a;'2  -  3x*  =  4, 
x2  =  4, 
x  =  ±  2. 

Note  1.— This  result  might  be  written  ±  x  —  ±  2  ;  but  this 
would  not  be  different  from  x  =  ±  2. 


QUADRATIC   EQUATIONS.  85 

Note  2. — If  we  have  x°  =  a,  and  a  is  not  a  perfect  square,  we  write 
x=  ±  Va. 


,.,      _     x        x*       x-        8 
Ex.  2.  _--T+-=T. 


Clearing  the  equation  of  fractions, 
4z'2  -  dx°-  +  x-  =  32, 
3a;2  =  32, 
x>  =  1G, 
x  =  ±  4. 

Examples — 37. 

Solve  the  following  pure  quadratics 

1.  9a;2  -  4  =  3z2  +  5. 

2.  2a;2  -  12  =  36  -  8a;2. 


4r2 
3.   ~  =  C25. 


iS-» 


a;  +  1       a;  —J.  _  13 
a?  -  1  +  aTTT  ~=  T 


a;-  -  1         a;-       %xl 

6'  -7~-T  =  X-23- 


7.   (3a?  +  1)  (3a;  -  1)  =  81a:2  -  33. 


86  SOLUTION  OF  AFFECTED   QUADRATICS, 


x-2  __  _30_ 

2      ~  x~+~2 


9.   (x  +  2)2  =  4:X  +  20. 


10.  * -£-  =  6 — - 


SECTION    XXVIII. 
SOLUTION  OF  AFFECTED  QUADRATICS. 

98.  To  solve  an  equation  which  has  both  x2  and  x  in  it, 
we  bring  it  to  a  simple  equation  by  taking  the  square  root. 
To  do  this  we  have  the  following 

Rule. — 1.  Reduce  the  equation  to  the  form  x2  +  px  =  q, 
in  which  x2  has  +  \  for  coefficient. 

2.  Then  add  the  square  of  half  the  coefficient  of  x  to  both 
sides;  thus  making  the  first  side  a  perfect  square,  and  pre- 
serving the  equality. 

3.  Take  the  square  root  of  both  sides,  putting  the  sign 
±  before  the  root  of  the  second  side.  Then  find  x  in  tli  is 
simple  equation. 

Ex.  1.  3z2  -  \%x  =  36. 

Dividing  by  o,  wc  have 

x-  -  Ax  —  12. 


SOLUTION  OF  AFFECTED  QUADRATICS.  87 

Adding  the  square  of  2  to  both  sides,  we  get 
x-  —  4ic  +  4  =  12  +  4, 
or  x-  —  4x  +  4  =  16. 

Taking  the  square  root  of  both  sides, 

x-2  =  ±4; 
whence,  a  =  2  +  4  =  6, 

or  a;  =  2  —  4  =  —  2. 

Note. — When  the  second  side  is  not  a  perfect   square,  we  put 
the  sign  ±  V       over  it. 

Ex.  2.  x°-  +  Gx  =  2. 

Adding  the  square  of  3  to  both  sides,  we  get 
x?  +  6x  +  9  =  11. 

Taking  the  square  root,       x  +  3  =  ±  Vll, 

whence,  x  =  —  3  -1-  Vll, 

or  ic=  — 3  —  VTT, 

and  we  can  only  find  x  approximately  by  getting  the  approximate 
square  root  of  11. 

Ex.  3.  Sx2  +  15s  =  18. 

Dividing  by  3,  x-  +  5z  =  6, 

x'  +  5x  +  *£  =  6  +  '*$■  =  449 

x  +  |.=  ±h 
x  =  —  1  +  1  =  1,     or    x~  —  f  —  |=—  6. 


88  SOLUTION  OF  AFFECTED  QUADRATICS. 

Ex.  4,  x*  +  -V-£  =  V3-. 

«2  +  ^  +  (f)2  =  ¥  +  ¥  =  ¥, 

x  +  $=  ±|, 
and  a>  =  -£  +  f=-l,     or    a;  =  -  |  -  §  =  -  ±3\ 

Ex.  5.  a2  -  \x  =  34. 

»2-^  +  (i)2  =  34+s^  =  -4F, 

*  -  i  =  ±  ¥, 
and  a;  =  £  +  ^  =  6,    or    x  =  £  —  \*  =  —  Y. 

Examples — 38. 

1.  2x*  -  12x  +  40  =  a2  +  Gx  -  5. 

x  —  Z       x  +  2         d 

3.  a2  -  4#  ==  5. 

4.  xl  —  3a;  =  2a;2  —  x  —  £. 

5.  a;2  —  Ga;  =  —  55. 

6.  x2  -x  =  42. 

7.  #  =  f #  —  ^j. 

8.  2a:2  -  9x  =  110. 

9.  100z2  +  80a;  =  9. 

10.  9z2  -  7x  =  16. 

11.  11a;2  -  3a;  =  14. 


PROBLEMS-QUADRATIC  EQUATIONS.  89 

x  +1        x         3 


13.  -JO     +_40    =9_ 

#  +  1       x  —  1 


u.  -1 L_  =  A 

x  +  3       36 


# 


3^+^_3^-2_ 
3a7-^~~  3*7+2         2* 


16.  — t-?  -  ^Zli  =  7. 
5  —  a;        6  —  x 


17.   (a;  -  2)2  +  (x  -  3)  (»-!)=«■-  8. 


SECTION     XXIX. 

PROBLEMS  GIVING   RISE  TO   QUADRATIC 
EQUATIONS. 

99.  Ex.  1.  Find  a  number  such  that  its  square  is  equal 
to  twice  the  product  of  two  numbers,  one  of  which  is 
greater  by  3,  and  the  other  less  by  4. 

Let  x  ==  the  number.  Then  x  +  3  =  the  number  greater  by  3,  and 
x  —  4  =  the  number  less  by  4. 

Then  x2  =  2  (x  +  3)  (x  -  4), 

that  is,  x2  =  2x2  -2x-  24, 

reducing,  x2  —  2x  =24, 

completing  the  square,  x2  —  2x  +  1  =  24  +  1  =  25, 
taking  the  square  root,  x  —  1—  ±5, 

whence  #  =  1  +  5  =  6,  or  a;  =  1  —  5  =  —  4. 


90  PROBLEMS— QUADRA  TIC  EQ UA  TIONS. 


Examples — 39. 

1.  Find  a  number  which  multiplied  by  its  excess  over 
21  gives   196. 

2.  Find  the  number  whose  square  increased  by  4  times 
the  number  is  equal  to  117. 

3.  Find  the  number,  5  times  the  square  of  which  di- 
vided by  3  is  135. 

4.  Find  the  number  whose  increase  by  60  multiplied  by 
its  excess  over  60  gives  6400. 

5.  Two  numbers  are  in  the  ratio  of  5  to  3,  and  the 
difference  of  their  squares  is  equal  to  144.  Find  the  num- 
bers. 

6.  Find  the  fraction  which  exceeds  its  square  by  f. 

7.  Find  two  numbers  the  difference  of  which  is  5,  and 
the  product  of  the  greater  by  their  sum  is  equal  to  150. 

8.  Find  two  consecutive  numbers  the  product  of  which 
is  5  times  the  sum  of  the  numbers  increased  by  5. 

9.  Find  a  number  such  that  12  divided  by  the  number, 
added  to  12  divided  by  the  number  increased  by  9,  is 
equal  to  5. 

10.  A  man  bought  $100  worth  of  sheep.  He  lost  5  of 
them,  and  sold  the  rest  for  $100,  and  gained  $1  a  head  on 
those  sold.     Find  the  number  of  sheep  which  he  bought. 


SIMULTANEOUS  EQUATIONS— QUADRATICS.        91 


SECTION    XXX. 

EASY  SIMULTANEOUS  EQUATIONS  SOLVED 
BY   QUADRATICS. 

100.  We  will  consider  the  case  of  two  equations  and  two 
unknowns,  when  one  of  the  equations  is  of  the  first  degree, 
or  a  simple  equation,  and  the  other  of  the  second  degree, 
or  a  quadratic  equation.     For  this  case  we  have  the 

Rule. — From  the  equation  of  the  first  degree  find  the 
expression  of  one  of  the  unknown  quantities  in  terms  of 
the  other,  and  then  substitute  this  expression  in  the  second 
equation. 

Ex.  1.  Given  x  +  y  =  10     (1)  )  to  find  the  values  of 
x1  +  if  =  52     (2)  )  x  and  y- 

From  (1),  y  =  10  -  x. 
Putting  this  in  (2),  x-  +  (10  -  xf  =  52. 
Expanding  (10  -  xf,  x2  +  100  -  20z  +  x1  -  52. 

Uniting  terms,  etc.,  2x-  —  20.r  =  —  48. 

Dividing  by  2,  x-  —  10^  =  —  24. 

Completing  square,  xlJ  —  10.E  +    25  =  —  24  +  25  —  1. 
Taking  the  square  root,    .  x  —      5  —  ±  1. 
Transposing,  x  =  6  or  4. 

Substituting  value  of  x  in  (1),  y  —  4  or  6. 


92        SIMULTANEOUS  EQUATIONS— QUADRATICS. 
Ex.  2.  5x  -  2ij  =  4     (1) 


to  find  the  values  of  x  and  y. 

dxy  -  4:X>  =2     (2)  ' 


From  (1),  x  =  i±Jl 


Putting  tins  for  a  in  (2),  3y  (—J^)  -  4  (— -J^)*=  3» 
12y  +  faf       (64  +  My  +  %2) 


-2. 


Reducing,  7y2  -2y  =  57, 


y      77 


2y  _57 

19 

and  y  —  o,   or — . 

The  first  value  of  y,  substituted  in  first  equation,  gives 
x  =  2. 

Examples— 40. 

Solve  the  simultaneous  equations 


X 

+  y 

;i' 

[I)- 

xy 

— 

6  ) 

X 

xl 

+  V 

= 

::s 

(2). 

x1 

X 

+  y 

:   100 

14 

j(3). 

SIMULTANEOUS   EQUATIONS— QUADRATICS.        93 

x  —    y  =  5 


Bx  +  4#  =?  2zy  -  12 


xy  =  432  ^1 


a; 

:3 

J 

xy  = 

5x  = 

:48C 
:      6^ 

(6). 
') 

a; 

5 
3 

1 

*y  = 

:   6 

J 

z2-</2  = 
x-y  = 

:57 
:3 

|(8)- 

%x  +  Qy  = 

Qx*-2y*  = 

:38; 

:4G 

|(9). 

vy  +  7y  = 

x-y  = 

:24 
:      3 

|  (10). 

x  +  ?/  =  7 

'  (11). 
2#  +  7/  ==  4r?/  —  2 


94  RADICALS  OF  THE  SECOND  DEGREE. 


SECTION    XXXI. 

RADICALS   OF   THE  SECOND  DEGREE. 

101.  Radicals  or  Surds. — The  indicated  roots  of 
quantities  which  are  not  perfect  powers  are  called  Radi- 
cal expressions,  or  simply  Radicals  or  Surds,  as,  for  ex- 
ample, ^/5a,  A^ll,  A^S,  3yi2,  which  interpreted  by 
signs  mean  that  \/ba  x  \/ha  =  5a  ;  \Al  x  V  H  x  a/H  = 
11;    \/8  x  a/8  x  a/8  x  ^8  =  8,  etc. 

102.  Radicals  of  the  Second  Degree.— The  indi- 
cated square  roots  of  quantities  which  are  not  perfect 
squares  are  called  radicals  of  the  second  degree,  as  5  Va, 
Via2,    \/%ab,  etc. 

103.  Coefficients. — The  factor  before  the  radical  sign 
is  called  the  coefficient  of  the  radical.  Thus  in  the  radi- 
cals 5Vffl,  QaVbc,  5  and  6a  are  the  coefficients  respect- 
ively.    When  no  coefficient  is  written,  1  is  understood. 

104.  1.  A  coefficient  of  a  radical  of  the  second  degree 
may  be  passed  under  the  radical  sign  as  a  factor  by  squar- 
ing it.     For,  since  the  V{5)2  is  5,  then  5\/#  =  \^5a  x  a 


Vs 


oa. 


2.  Any  perfect  square  factor  of  an  expression  under  the 
radical  sign  may  be  transferred  as  a  factor  before  the 
sign  by  taking  its  square  root. 


Thus,    v%5a?b  —  V(5ay  x  b  =  ha\Zb, 

V45  =  7v'9~x~5~=  7  x  3\/5~=  21  y/Si 


RADICALS   OF  THE  SECOND  DEGREE.  95 

105.  Simplest  Form  of  Radicals.— A  radical  quan- 
tity of  the  second  degree  is  in  its  simplest  form  when  the 
number  or  expression  under  the  radical  sign  contains  no 
factor  greater  than  1  which  is  a  perfect  square. 

Thus,  5V3  is  in  its  simplest  form,  but  %^/%a*  is  not 
in  its  simplest  form,  since  it  may  be  written  2aA#2  x  2, 
and  taking  the  square  root  of  the  factor  4#*2,  we  have 
2  V&f  =  2\/4aa  x  2  =  4«a/2. 

106.  Reduction  of  Radicals. — To  reduce  a  radical 
of  the  second  degree  to  its  simplest  form,  we  have  the  fol- 
lowing 

Rule. — Separate  the  part  under  the  radical  sign  into 
two  factors,  one  of  which  contains  all  the  factors  which  are 
perfect  squares.  Take  the  square  root  of  this  factor  and 
multiply  it  ly  the  coefficient  of  the  radical,  leaving  the 
other  factor  under  the  radical  sign. 

Ex.  1.  Reduce  Vo4  to  its  simplest  form. 
Reduction,  VM  =  V¥V§  =  dVW. 

Ex.  2.  Reduce  ^WlcrWc  to  its  simplest  form.  / 

Reduction,  2  V27aWc  =  2  Vdtfb-  x  dao  =  2  x  dab  Vo~ac  = 

Gab  V'dac. 

107.  Similar  Radicals.— Radicals  of  the  second  de- 
gree are  said  to  be  similar  when  they  have  the  same  quan- 
tities under  the  radical  sign.  Thus  5 a/3,  8\/3,  10y3 
are  similar  radicals,  and  <s/\%  may  be  rendered  similar  to 
all  these  because  in  its  simplest  form  *J\%  =  %\/3. 


96 


RADICALS  OF  THE  SECOND  DEGREE. 


108.  Addition  and  Subtraction  of  Radicals. — 

To  add  or  subtract  similar  radicals, 

Rule. — Add  or  subtract  their  coefficients,  and  place  the 
result  as  a  coefficient  before  the  common  radical. 

Ex.  1.  6V«  +  12a/o~+  23 V«  =  41  Va. 

Ex.  2.  W^a  ~  3V&a  =  V~5a- 

Ex.  3.  Add  3  a/54,    5  a/24,  and  3  a/6. 

Here    3  VM =  3  V9~x~6  =  9  Vft,    5  VU  =  5  VZ1T6  =  10  ^ 5 

and  9  V6  +  10  V6"+  3  ^6  =  22  ^ 

109.  Multiplication  and  Division  cf  Radicals.— 

To  multiply  or  divide  two  radicals  of  the  second  degree, 

Rule. — Multiply  or  divide  the  coefficients  for  the  new 
coefficient,  and  the  parts  under  the  radical  for  the  new 
radical  factor. 


Ex.  1.  Multiply  5V7  by  8 a/3. 
Ex.  2.  Divide  9aW%b  by  3aV%- 
Ex.  3.  Square  3  4-  a/5. 

Process,     3  +  ^5 

3  +  VW 


Result,  40 a/21. 
Result  3#a/#- 


9  +  3  V'5 
3V5  +  5 


Result,     9  +  6  ^5  +  5 


RADICALS  OF  TEE  SECOND  DEGREE.  97 

Ex.  4.  Multiply  3  4-  V%  by  3  -  V*. 

(3  +  V2)  (3-  VS)  =  (3)2  -  (V2)2  =  9-2  =  7. 

Note. — The  above  rules  for  the  addition,  subtraction,  multiplica- 
tion, and  division  of  radicals  of  the  second  degree  apply  to  all  radicals 
of  the  same  index  or  degree. 

_ 

110.  Radical  Expressions  of  the  Form  y  -j-.  I'rac- 

/ 
tional  radical  expressions  of  the  form  y  — ,  are  simplified 

by  the  following 

Rule. — Multiply  the  numerator  and  denominator  of  the 
fraction  by  any  number  which  loill  make  the  denominator 
a  perfect  square  ;  then  apply  the  Rule  of  Art.  104. 

Thus  |/t=  </%  =  j/*   x  oft  =  IVab. 

boo  o 

Ex.   Reduce  oy^to  its  simplest  form. 

5\/|  =  5a/|  =  oVi-  x  6  =  -fA/oV 


111.  Radical  Expressions  of  the  Form =. 

a  ±  Vb 

In  order  to  remove  the  radical  quantity  from  the  denomi- 

c                       c 
nator  of  such  expressions  as =   or  — ,    we 

a  +  s  h  a  —  y'j 

multiply  the  numerator  ami  denominator  of  the  first  by 
a  —  \/b.  and  of  the  second  by  a  +  Vb. 


98  RADICALS  OF  THE  SECOND  DEGREE. 

c  c(a  —  Vb )  _ac  —  cVb 


Thus 


and 


Similarly 


a  +  Vb         (a  +Vb)  (a  —  Vb  )  a2  —  b 

c  _  c(a  +Vb)  _  ac  +  c\/b 

a-Vb  ~  (a-Vb)  (a+Vb)~       a2  -  b 

1  2  -a/3  2  -  a/3 


2+a/3  (2  +  a/3)   (2 -a/3)  4-3 

2  -V3. 

Examples — 41. 

1.  Simplify  \/28«W. 

2.  Simplify  a/#3  —  «^2. 

3.  Reduce  a/27,  a/12,  and  5  a/48  to  similar  radicals. 

4.  Reduce  a/8,  a/50,  and  a/72  to  similar  radicals. 

5.  Add  a/2,  5  a/8-,  4a/18,  3a/128. 

6.  Multiply  3 a/2  -  5 a/3  by  2 a/2. 

7.  Multiply  2  +  a/3  by  3  -  a/3. 

8.  Multiply  \/a  +  b  by  \/a  —  b. 

9.  Multiply  a/5  +  a/3  by  a/5  -  a/3. 

10.  Square  2  -  a/3. 

11.  Simplify  Vf 

12.  Simplify  f  Vf 

13.  Divide  2 a/3  by  3 a/5  and  simplify  the  result. 


RADICALS  OF  TEE  SECOND  DEGREE.  99 

3 

14.  Remove  the  radical  from  the  denominator  of — = . 

V2  -1 

o 

15.  Eemove  the  radical  from  the  denominator  of 


V3+1 

16.   Remove   the    radicals    from    the    denominator    of 


V5  ~  V% 
17.  Simplify  ^80  +  3^/20  -  4^45  +  2a/5. 

EQUATIONS    CONTAINING    RADICALS    OF 
THE    SECOND    DEGREE. 

112.  When  the  unknown  quantity  in  an  equation  is  un- 
der a  radical  sign,  it  becomes  necessary  in  solving  it  to 
clear  the  equation  of  radicals.  If  the  radicals  are  of  the 
second  degree,  and  if  there  be  only  one  in  the  equation, 
we  have  the  following 

Rule. — Transpose  the  terms  so  that  the  radical  shall  he 
on  one  side  of  the  equation  a?id  all  the  other  terms  on  the 
other  side.     Then  square  loth  sides. 


Example. — Given  V*>  +  #  +  8  =  14,  to  find  x. 


Transposing,  V6  +  x  =  6. 

Squaring  both  sides,  6  +  x  =  36  ; 

and  x  =  30. 

113.  If  the  equation  contains  two  radicals  of  the  second 
degree  with  the  unknown  letter  under  them,  we  must  re- 
peat the  above  operation  for  the  second  radical ;  that  is, 
two  transpositions  and  two  squarings  will  be  necessary. 


100  RADICALS  OF  THE  SECOND  DEGREE. 


Example.— Given  <s/x  +  6  +  V%  —  5  =  11,  to  find  a. 


Transposing,  Va  +  6  =  11  -  Vx  -  5. 


Squaring,  *  +  6  =  (11  -V  x  -  5)9 


a;  +  6  =  121  -  22  Vz  -  5  +  x  -  5. 
Transposing  and  reducing, 


22  Vx  -  5  =  110  ; 

or 

Squaring  again, 
and 

Yx  -  5  =  5. 
re  -  5  =  25  ; 
re  =  30. 

Examples — 42. 
Find  x  in  the  following  equations 


!•   \/a  +  x  +  b  =  c. 

2.  V#  +  25  —  \Z«  =  1- 

3.  aA  +  13"—  Vx  —  11  =  3. 


4.   V#  +  4  +  V^  —  1  =  5- 


5.   V*x  +  14  +  V&z  -14=  14. 


6.   V4z  +  4  +  9  =  13. 


7.   y#  +  4  +  V%  —  4  =  4. 


8.  a/5z  -  1  +  \A0a  +5  =  8. 

9.  V%  +  9  =  13  -  2V«. 


RATIO  AND  PROPORTION.  101 

SECTION     XXXII. 
RATIO   AND    PROPORTION. 

114.  Ratio  is  the  relation  which  one  quantity  bears  to 
another  in  respect  of  magnitude,  and  is  measured  by  the 
number  of  times  the  one  is  contained  in  the  other,  or  by  the 
part  or  parts  the  one  is  of  the  other. 

Thus,  the  ratio  of  10  to  5  is  2,  and  the  ratio  of  5  to  10 
is  y5^,  or  \,  as  5  is  one-half  of  10. 

115.  Hence  the  fraction  -j-  represents  the  ratio  of  a  to 

b.     This  ratio  is  written  a  :  b.     Hence,   a  :  b  =  -j- ,  and 

•  c 

similarly  c  :  d  =  -=- ,  etc. 

a         c 

116.  Proportion.— If  -=-  =  -r ,  a  :  b  =  c  :  d,   and 

this  equality  of  two  ratios  is  called  a  Proportion.  It  is 
usually  written  a  :  b  :  :  c  :  d,  and  is  read  "a  is  to  b  as  c 
is  to  d." 

The  first  and  third  terms  of  the  proportion  are  called 
the  antecedents,  and  the  second  and  fourth  the  consequents. 
Again,  the  first  and  fourth  terms  are  called  the  extremes, 
and  the  second  and  third  the  means. 

117.  Results  to  be  Remembered.— We  must  bear 
in  mind  : 

1.  The  value  of  any  ratio  a  :  b  is  the  fraction  -=- . 

2.  If  a  :  b  :  :  c  :  d,  then  -^-  =  -=■ , 

b         d 


102  RATIO  AND  PROPORTION. 

3.  Then,  the  rules  given  for  fractions  and   equations 
apply  to  ratios  and  proportions. 

118.  If  the  terms  of  a  ratio  be  multiplied  or  divided  by 

the  same  number  the  ratio  is  not  changed.    For  -j-  =  —7 , 

and  therefore  a  :  b  :  :  ma  :  ml). 


Examples — 43. 

Find  the  values  of  the  following  ratios : 

1.  3x  :  %lx  ;         am  :  mn. 

2.  2a"  :  Sa* ;        cxy  :  dy. 

3.  ac  :  be  ;        hacx  :  £a2x. 

4.  bcx2  :  bacx  ;        4«V  :  12aV. 

5.  ax  +  bx  :  2x2  ;        2ab  +  b~  :  be. 

6.  1  —  a2  :  1  +  a  ;.        a2  —  Xs1  :  a  —  x. 

7.  \bc  :  \ac  ;        \amn  :  -g^aiv. 

8.  Which  is  the  greater  16  :  17,  or  17  :  18. 

PROPORTION. 

119.  In  any  proportion  the  product  of  the  extremes  is 

equal  to  the  product  of  the  means. 

For  let  a  :  b  :  :  c  :  d  be  the  given  proportion. 

a         c 
Then  j-.  =  -y,  and  clearing  of  fractions,  ad  —  be,  which 

was  to  be  proved. 

It  follows  that  if  a  :  b  :  :  b  :  e,  then  b2  =  ac,  or  b  —  Vac  ; 
b  is  then  said  to  be  a  mean  proportional  between  a  and  c 


RATIO  AND  PROPORTION.  103 

Hence,  Rule. — The  mean  proportional  between  two  num- 
bers is  the  square  root  of  their  product. 

120.  Conversely,  if  ad  —  be,  then  the  proportion  a:  b  :  : 
c  :  d,  is  true. 

For,  dividing  both  sides  of  the  equation  ad  =  be  by  bd, 

a        c 
we   have  -j-  =  -j .     Hence  (Art.  117),  a  :  b  : :  c:da 

121.  Proof  of  the  Single  Rule  of  Three.— It  fol- 
lows from  the  above,  if  three  terms  of  a  proportion  are 
given,  the  fourth  may  be  found. 

For,  let  the  given  terms  be  a,  b,  c,  in  order,  and  the  un- 
known term  be  x ;  then  a  :  b  :  :  c  :  x,  and  therefore  ax  —  be, 

be 
or  x  =  —  .     Hence. the  rule  as  given  in  the  arithmetic. 

122.  If  a  :  b  :  :c  :d,  then  b  :  a  :  :  d :  c. 

For,  since  a  :b  :  :c  \d,  then  be  =  ad  (Art.  119) ;  and  di- 
viding both  sides  by  ac,  we  have  —  =  — ,  or 

a         c 

b  :  a  :  :  d  :  e. 

123.  If  a :  b  : :  c  :  d,  then  a  :  c  :  :  b  :  d. 

For,  ad  =  be  (Art.  119) ;  and  dividing  both  sides  of  this 

equation  by  dc,  we  have  —  —  ~tj  or 

c         a 

a  :  c  :  :  b  :  d. 

124.  If  a  :  b  :  :  c  :  d ',  then  a  -j-  b  :  b  :  :  c  +  d :  d. 

For,  first,  —  =  — ,  and  adding  1  to  both  sides  of  this 

equation,  we  have    -=-  +  1  =  -=-  +  1. 

o  a 


104  RATIO  AND  PROPORTION'. 

Hence  a-±*  -  -+-  • 

nence,  b     _     d    , 

.*.     a  +  b  :  b  :  :  c  +  d  :  d. 

Also,  since  —  =  — ,  it  is  clear  that 
a        c 

a  +  b  :  a  :  :  c  +  d  :  c. 

125.  If  a  :  b  :  :  c  :  d,  then  a  —  b  :  b  :  :  c  —  d  :  d. 

(I  c 

First,  -  -  —  — ,    and  subtracting  1  from  both  sides  of 

this  equation,  we  have 


or 


«  —  #        c  —  d 

b               d      ' 

Hence, 

a  —  b  :  b  :  :  c  —  d  :  d. 

Also, 

a  —  b  :  a  : :  c  —  d  :  c. 

126.  It  a  :b  wed   (1),    and   a:b  :  :e  :f  (2),    then 
c  :  d  :  :  e  :  /. 

For,  first, 
and  second, 


y 

-~d' 

"5" 

e 

=  7' 

y 

e 

=  7' 

Hence, 

Therefore,  a  :  b  :  :  e  :f. 


RATIO   AND   PROPORTION.  105 

Examples — 44. 

1.  If  a  :  b  : :  c  :  d,  show  that  5a  :  6b  : :  5c  :  6d. 

2.  Change  5  :  #  +  3  :  :  x  —  3  :  8  into  an  equation,  and 
find  £. 

3.  Change  4#  =  5y  into  a  proportion. 

4.  If  2  —  a;  :  2  +  x  :  :  5  :  6,  find  the  ratio  2  :  x. 

5.  The  first,  second,  and  fourth  terms  of  a  proportion 

are  ax,  3cx,  and — ,  what  is  the  third  term  ? 

a 

6.  There  are  two  numbers  in  the  ratio  of  2  to  3,  and  if 
each  of  these  be  increased  by  2  the  ratio  is  then  5  ta  7. 
Find  the  two  numbers. 

7.  If  6a  -  4c  =  4fZ  -  6b,  show  that  a  +  b  :  c  +  d :  :2  :3. 

8.  Find  two  numbers  in  the  ratio  of  4  to  5,  the  differ- 
ence of  which  bears  to  54  the  ratio  of  1  to  9. 

9.  Given  x  +  y  :x  —  y\  :  6  :  5.     Find  the  ratio  x  :  y. 

10.  Find  the  mean  proportional  between  16  —  x*  and 
4  +  x1 

4  —  x~ ' 

11.  Change  a?  +  4:4::12:#  +  6  into  an  equation, 
and  find  x. 

12.  There  are  four  consecutive  numbers,  the  last    of 
which  bears  to  the  first  the  ratio  of  8  to  7.     Find  them. 

5* 


IOC  ARITHMETICAL    PROGRESSION. 

SECTION     XXXIII. 
ARITHMETICAL   PROGRESSION. 

127.  An  Arithmetical  Progression  is  a  series  of 
numbers  which  go  on  increasing  or  decreasing  by  a  fixed 
number  called  the  Common  Difference. 

Thus  1,  3,  5,  7,  9,  11,  etc.,  form  an  arithmetical  pro- 
gression with  the  common  difference  2. 

So  also  20,  17,  14,  11,  8,  etc.,  is  an  arithmetical  pro- 
gression with  the  common  difference  3. 

Again,  a,  ka,  la,  10a,  etc.,  is  an  arithmetical  progres- 
sion with  the  common  difference  3a. 

128.  General  Forms  of  Arithmetical  Progres- 
sions.— If  a  be  the  first  term  of  an  arithmetical  progres- 
sion, and  d  the  common  difference,  then  the  general  alge- 
braic forms  will  be 

1.  a,  a  +  d,  a  +  2d,  a  +  3d,  etc.,  for  an  increasing 
arithmetical  progression. 

2.  a,  a  —  d,  a  —  2d,  a  —  3d,  etc.,  for  a  decreasing 
arithmetical  progression. 

129.  To  find  any  Term. — From  these  we  see  that 
given  the  first  term  a,  and  common  difference  d,  of  an  arith- 
metical progression,  we  can  find  any  term. 

Thus  (1.)  in  an  increasing  arithmetical  progression  the 
6th  term  is  a  +  bd,  the  10th  term  is  a  +  Oaf,  and  the  wth 
term  is  a  +  (n  —  l)d. 


ARITHMETICAL  PROGRESSION.  107 

(2.)  In  a  decreasing  arithmetical  progression  the  nth 
term  is  a  —  (#t  —  l)d. 

Therefore  to  find  any  term  of  an  arithmetical  progres- 
sion, we  have  the 

Rule. — Multiply  the  common  difference  by  the  number 
of  terms  which  precede  the  required  term,  and  add  this 
product  to  the  first  term  for  an  increasing  progression, 
and  subtract  it  from  the  first  term  for  a  decreasing  pro- 


130.  If  I  be  the  nth  term,  this  rule  may  be  expressed 
briefly  as  follows  : 

I  =  a  -f  (n  —  l)d,  for   an  increasing  arithmetical  pro- 
gression. 

I  =  a  —  (n  —  l)d,    for  a  decreasing    arithmetical    pro- 
gression. 

Ex.  1.   Find  the  25th  term  of  the  arithmetical  progres- 
sion 3,  5,  7,  9,  etc. 

Here  a  —  3,     and     d  =  2,     and     n  —  25. 
Hence  I  =  3  +  (25  -  1)2  =  3  +  48  =  51. 

Ex.  2.  Find  the   20th  term    of   the  arithmetical    pro- 
gression 80,  76,  72,  68,  etc. 

Here  a  =  80,     d  =  4,     and     n  =  20. 

Hence        I  =  80  -  (20  -  1)4  =  80  -  76  =  4. 


108  ARITHMETICAL  PROGRESSION. 

131.  Sum  of  the  Terms.— To  find  a  rule  for  the  sum 
of  n  terms  of  an  arithmetical  progression,  we  write  first 

S=  a  +  a  +  d  +  a  +  2d  +  a  +  3d,  etc.,  to n terms; 

and  also,  S  =  1  -\-l  —  d  +  I  —  2d  +  I  — 3d,  etc.,  to  n  terms. 


Adding,  we  have  2S  =  (a  +  I)  +  (a  4-  t)  +  (a  +  T)  +  {a  +  I)  to  n  terms, 
or  2S—  (a  +  l)n. 


Therefore  S  —  I ]  i 


Hence,  Rule. — Multiply  half  the  sum  of  the  first  and 
last  terms  by  the  number  of  terms. 

Example. — Find  the  sum  of  30  terms  of  the  arithmeti- 
cal progression  1,  3,  5,  7,  9,  etc. 

Here  a  =  1,     d  =  2,     and     n  =  30. 

Hence,  first,    I  =  1  +  (30  -  1)  2  =  1  4-  58  =  59, 

and,  second,        S  =     "t        x  30  ==  30  x  30  =.900. 

132.  Arithmetical  Means. — Numbers  inserted  be- 
tween two  given  numbers,  and  forming  an  arithmetical  pro- 
gression with  the  given  numbers  as  first  and  last  terms,  are 
called  arithmetical  means. 

To  find  a  rule  for  inserting  any  number  of  arithmetical 
means  between  two  numbers  a  and  b,  we  have  only  to  find 


ARITHMETICAL  PROGRESSION.  109 

the  common  difference.     To  do  this,  we  take  the  formula, 
Art.  130, 

I  =  a  +  (n  —  l)d, 

1    _    n 

whence  d  — 


n-1 


Now  if  m  be  the  number  of  means  to  be  inserted,  be- 
tween a  and  I,  then  n  =  m  +  2,  and  »  —  1  =  m  +  1. 

Therefore  d=—  -— ,  which  gives 
m  +  1 '  h 

Rule. —  To  find  the  required  common  difference,  divide 
the  difference  letween  the  numbers  by  the  number  of  means 
to  be  inserted  plus  1. 

Ex.  1.  Insert  4  arithmetical  means  between  5  and  20. 

20 5       25 

Here  the  common  difference  d  =  —. T  =  -=-  =  3. 

4+1        5 

Hence  the  four  required  means  are  8,  11,  14,  17,  and 
the  arithmetical  progression  is  5,  8,  11,  14,  17,  20. 

133.  To  find  an  arithmetical  mean  between  two  num- 
bers a  and  I,  or  their  average,  let  x  =  this  average  or 
arithmetical  mean,  and  d  the  common  difference. 

Then  x  =  a  4-  d, 

also,  x  =  I  —  d, 

and  adding,  2x  =  #  +  /, 

a  +  / 


or 


2 


Hence,  Rule. — 77>e  arithmetical  mean  or  average  of 
ttvo  numbers  is  half  their  sum. 


110  ARITHMETICAL   PROGRESSION. 

Examples — 45. 

1.  Find  the  14th  and  28th  terms  in  the  progressions  : 

(1.)  3,  7,  11,  etc. 
(2.)  5,  8,  11,  14,  etc. 

\°')    ¥>    t>    2>   e^C* 

(4.)  200,  105,  190,  etc. 

2.  Find  the  sum  of  20  terms  of  the  progressions  : 

(1.)  1,  4,  7,  10,  etc. 
(2.)  100,  96,  92,  etc. 
(3.)  15,  14|,14i,  etc. 

3.  The  clocks  of  Venice  strike  the  hours  from  1  to  24. 
Find  the  number  of  strokes  in  a  day. 

4.  A  man  employed  a  workman  for  90  days  at  1  cent  the 
first  day,  3  cents  the  next,  5  cents  the  next,  etc.  Find  the 
whole  amount  of  the  workman's  wages  for  the  90  days. 

5.  In  a  potato  race  each  man  picked  up  50  potatoes 
placed  two  feet  apart  in  a  line,  and  put  them  in  a  basket  in 
the  line  at  two  feet  from  the  first  potato.  How  far  did 
each  run,  starting  at  a  basket  ? 

6.  Show  that  the  sum  of  50  terms  of  the  arithmetical  pro- 
gression 1,  3, 5,  7,  9,  etc.,  is  50',  and  the  sum  of  n  terms  is  n\ 

7.  Show  that  the  sum  of  the  first  n  natural  numbers 

1,  2,  3,  4,  5,  6,  7,  8,  etc.,  is        ^      . 

8.  Find  the  arithmetical  mean  of  \  and  \, 


GEOMETRICAL  PROGRESSION.  Ill 

9.  Find  the  arithmetical  mean  of  a  +  b  and  a  —  b. 

10.  Insert  two  arithmetical  means  between  10  and  11. 

11.  Insert  4  arithmetical  means  between  50  and  40. 

12.  Insert  20  arithmetical  means  between  3^  and  3|. 

13.  A  heavy  body  falling  from  a  height  falls  16T12  feet 
the  first  second  of  its  fall,  and  in  each  succeeding  second 
32|  more  feet  than  in  the  preceding  (the  resistance  of  the 
air  being  left  out).  How  far  would  a  body  fall  in  20 
seconds  ? 


SECTION    XXXIV. 
GEOMETRICAL    PROGRESSION. 

134.  A  Geometrical  Progression  is  a  series  of 
numbers  which  go  on  increasing  or  decreasing  in  the  same 
fixed  ratio,  that  is,  by  a  common  multiplier. 

Thus,  2,  4,  8,  16,  etc.,  are  in  geometrical  progression, 
each  term  being  twice  the  preceding  term. 

Also,  81,  27,  9,  3,  1,  are  in  geometrical  progression, 
each  term  being  ^  of  the  preceding  term. 

135.  This  common  multiplier  is  called  the  Common 
Ratio,  and  may  be  whole  or  fractional.  It  is  found  in  any 
geometrical  progression  by  dividing  any  term  by  the  pre- 
term. 


Ex.  1.  The  common  ratio  in  the  geometrical  progres- 
sion 3,  9,  27,  81,  etc.,  is  -f  =  3. 


112  GEOMETRICAL  PROGRESSION. 

Ex.  2.  The  common  ratio  in  the  geometrical  progres- 
sion 64,  16,  4,  1,  is  \. 

Ex.  3.  Are  {,  \}  and  -fa   in   geometrical   progression  ? 

And  if  so,  what  is  the  common  ratio  ? 

Ans.  \  -r- \  =  %,  and  fa  -r  ^  =  f .  Hence  the  given  fac- 
tors are  in  geometrical  progression,  and  ^  is  the  common 
ratio. 

136.  General  Form  of  a  Geometrical  Progres- 
sion.— If  a  is  the  first  term  of  a  geometrical  progression, 
and  r  the  common  ratio, 

The  second  term  is  ar, 
The  third  term  is  ar'2, 
The  fourth  term  ar3, 

and  so  on,  the  exponent  of  r,  in  any  term,  being  always  1 
less  than  the  number  of  the  terms.  Therefore,  the  nth 
term  will  be  I  =  arn~  \  and  the  series  expressed  in  alge- 
braic form  will  be 

a,     ar,     ar1,     ar3,     ar*  ....  ar*-1. 

137.  To  Find  any  Term. — The  expression  I  =  arn~x, 
gives  us,  for  finding  any  term  of  a  geometrical  progres- 
sion when  the  first  term  and  common  ratio  are  known, 
the  following 

Rule. — Multiply  the  first  term  by  the  ratio  raised  to  a 
power  one  less  than  the  number  of  terms. 


GEOMETRICAL  PROGRESSION.  113 

Example. — Find  the  10th  term  of  the  geometrical  pro- 
gression 2,  4,  8,  16,  etc. 

I  =  2  x  29  =  1024. 

138.  The  Sum  of  the  Terms  of  a  Geometrical 
Progression. — Let  S  be  the  sum  of  the  terms  of  a  geo- 
metrical progression  a,  b,  c,  d,  e,  f  •  •  •  •  h,  I,  and  r  the 
common  ratio,  then 

I  =  ar, 

c  =  hr, 
d  =  cry 

I  —  kr, 


or  adding,  b  +  e  4-  d  •  •  •  +  I  =  (a  +  b  +  c  +  •  •  .  +  h  )r. 

But  the  first  side  has  all  the  terms  in  it  except  a,  and  the 

brackets,  on  the  second  side,  contain  all  the  terms  except  /. 

Hence, 

8  -  a  =  (8  -  l)r, 

or  S  —  a  =  Sr  -  Ir. 

Ir  —  a 


Therefore,  S  = 


r  -1 


This  expression  gives  S  when  r,  I,  and  a  are  known.    If 
the  geometrical  progression  is  a  decreasing  one,  then  the 

yalue  of  S  is  written   S  = . 

r  —  1 

Example. — Find  the  sum  of  11  terms  of  the  geometrical 

progression  2,  4,  8,  etc. 

Here  a  =  2,     r  =  2,     1  =  2x2'°  =  2048. 

„                                      „       2x  2048  - 1 
Hence,  S  = „ -. =  4094. 


114  GEOMETRICAL  PROGRESSION. 

{£    ^f 

139.  In  the  expression  S  =  3 —  for  a  decreasing  pro- 
gression, when  the  number  of  terms  increases  indefinitely 
(i.  e.,  becomes  great  without  limit),  Ir  becomes  a  smaller 

and  smaller  fraction,  and  if  we  neglect  it  we  get  — - — 

as  a  quantity  greater  than  the  sum  of  the  series  when  the 
number  of  the  terms  is  infinite. 

We  will  call  this  2  (sigma),  the  limit  of  the  sum  of  the 
terms  of  a  decreasing  geometrical  progression  when  the 
number  of  terms  is  great  without  limit,  and  write  it 

1  —  r 

Ex.  1.  Find  the  limit  of  the  sum  of  the  terms  of  the 
geometrical  progression  1,  J,  %,  i,  etc.,  for  an  infinite 
number  of  terms. 

a  11 

This  result  shows  that  the  sum  of  all  the  numbers  that 
can  be  written  in  the  geometrical  progression  1,  J,  J,  etc., 
can  never  be  as  great  as  2. 

Ex.  2.  Find  the  limit  of  the  sum  of  the  terms  of  the 
geometrical  progression  T\,  Tf^  toVo*  T.rihnr>  etc.,  or,  in 
other  terms,  the  recurring  decimal  .333333,  etc. 

Here  a  =  Ah     and    r  =  -rV. 

.a  .3.. 

Hence,  2  —  +  _  L  =  *x  ==  I  =  i- 


GEOMETRICAL  PROGRESSION.  115 

140.  Example. — Insert  a  geometric  mean  between  two 

numbers  a  and  b.     Let  x  be  the  mean.     Then  the  com- 

x  u 

mon  ratio  =  — ,  and  also  =  — . 
a  x 

Therefore,        —  —  — ,    or    x°-  =  ab  ; 
a        x 

and  x  =  Vab.     Same  result  as  in  Art.  119 

in  Proportion. 

141.  Example. — Insert  two  geometric  means  between  J 
and  2.     Let  x  =  the  required  common  ratio. 

Then  the  series  will  be 

4>     4^>     1%  >     "• 

Therefore  the  common  ratio 

2 

j 

4 

whence  x*  =  8. 


x  —  , 
±x- 


Therefore,  x  =  2, 

and  the  series  is  £,  £-,  1,  2. 

Examples — 46. 

1.  Find  the  common  ratio  in  the  following  geometrical 
progressions  : 

(1.)  100,     300,     000,  etc. 
(2.)  3J,     7,     14,  etc. 
(3.)  h     h     b  etc. 


216  GEOMETRICAL  PROGRESSION, 

(4.)  .2,     .02,     .002,  etc. 
(5.)  .625,     1.25,     2.5,  etc. 
(6.)  a,     Sax,     9ax-,  etc. 

2.  Find  the  geometric  mean  between  50  and  12^. 

3.  Insert  two  geometric  means  between  6  and  1G2. 

4.  Insert  two  geometric  means  between  ^  and  1. 

5.  Which  is  greater,  the  arithmetical  mean  or  the  geo- 
metric mean  between  1  and  ^  ? 

6.  Find  the  sum  of  10  terms  of  the  geometrical  pro- 
gression 1,  3,  9,  27,  etc. 

7.  A  blacksmith  used  eight  nails  in  putting  a  shoe  upon 
a  horse's  foot  ;  he  received  1  cent  for  the  first  nail,  2  cents 
for  the  second,  4  for  the  third,  and  so  on.  What  did  he 
receive  for  the  shoeing  ? 

8.  Find  the  limit  of  the  sum  of  an  infinite  number  of 
terms  of  the  geometrical  progression  J,  -fa,  -g^,  etc. 

9.  Find  the  limit  of  the  value  of  the  circulating  de- 
cimal .55555  

10.  Find  the  limit  of  the  value  of  the  decimal  .212121  •  •  • 

11.  Insert  three  geometric  means  between  12  and  192. 


MISCELLANEOUS  EXAMPLES.  117 

SECTION    XXXV. 
MISCELLANEOUS   EXAMPLES. 

142.  The  following  examples  may  be  used  according  to 
the  preference  of  the  teacher,  either  for  review  and  exam- 
ination after  the  pupil  has  accomplished  the  preceding 
sections,  or  may  be  drawn  on  for  additional  practice  while 
advancing  from  section  to  section. 

(Sections    I-VIII.) 

1.  Simplify  2a*x  —  3axi  +  x*  —  hcfx  +  6ax"  —  3x\ 

2.  Subtract  3a*b  -  2a2  +  6  -  Sx  from   -  3a2b  +  2a2 

+  6  -  8a;. 

3.  Add  a  —  b  +  c,     %a  +  2b  —  3c  —  da  +  45  +  4c, 

and  subtract  the  sum  from  4#  +  5b  —  be. 

4.  Multiply  2a  -  36  by  2«  +  3b. 

5.  Multiply  3«  +  b  by  3a  —  b. 

6.  Multiply  a  +  b  +  c  +  d  by  a  +  b  —  c  —  d. 

7.  Multiply  a2  +  a£-  +  62  by  a?  —  ab  +  b\ 

8.  Write  (5a  +  100b)9  and  (14«  -  11&)2. 

9.  Divide   2ba*x1y'    by    5ax*y,    and   the   quotient   by 

5aV#. 

10.  Divide  210«V  -  140«V  by  35a3#. 

11.  Divide  81a;4  —  if  by  3#  —  y. 

12.  Divide  a6  —  1  by  a3  +  2a2  +  2a  +  1. 


118  MISCELLANEOUS  EXAMPLES. 

13.  Divide  x*  —  4=a3x  +  3a*  by  x~  —  2ax  +  «2. 

14.  Simplify  (a  -  x)  -  (2x  -  b)  -  (2  -  2a)  +  (3  -  2*) 

-(I-*)- 

15.  Simplify  (x*  -  2cz2  +  3c*x)  -  (ex2  —  2x3  -J-   2cx2) 

+  (x3  —  c"x  —  ex'2). 

16.  Simplify  6(1  -  x)  +  2(1  +  6a). 

17.  Simplify  (1  +  a)  (1  —  x)  (1  +  a2). 

18.  Simplify  \(a  +  x)  —  \{a  —  a). 

19.  Find  the  factors  of  «4  —  b\ 

20.  Find  the  factors  of  x'  -  \\x  +  30. 

21.  Find  the  factors  of  »2  —  a  -  30. 

22.  Find  the  factors  of  4ra2  -  $n\ 

23.  Find  the  factors  of  16a2 x2  —  2by'. 

24.  Find  the  factors  of  x"  -  %x  +  14. 

25.  Find  the  factors  of  x'  —  Gz  —  7. 

2G.  Two   factors     of  b*  —  75  +  6  are  b  —  1  and  b—2. 
Find  the  other. 

(Sections   IX,    X.) 

27.  Find  the  G.  C.  D.  of  Wcx,  2a:x\  and  7acx. 

28.  Find  the  G.  C.  D.  of  a'  -  ¥  and  (a  -  b)\ 

29.  Find  the  G.  C.  D.  of  x*  +  x  —  6  and  a?  +  5z  +  6. 

30.  Find  the  G.  C.  D.  of  «2+  3a  +  2  and  a2  +  4a  +  3. 

31.  Find  the  G.  C.  D.  of  3a2  +  a  -  2  and  3a2  +  4a  -  4. 

32.  Find  the  L.  C.  M.  of  lx\  2\x\  63z7. 


MISCELLANEOUS   EXAMPLES.  119 

33.  Find  the  L.  C.  M.  of  14an,  63a2#,  2%ax\  and  70a;3. 

34.  Find  the  L.  C.  M.  of  UQaVc  and  221aW. 

35.  Find  the  L.  C.  M.  of  9(x  +  1)  and  6(x  +  2). 

36.  Find  the  L.  C.  M.  of  30(z  -  1)  and  A5(x  +  1). 

37.  Find  the  L.  C.  M.  of  6(x  +  a),     12 (x  -  a),     and 

18(z2  -  a2). 

38.  Find  the  L.  C.  M.  of  a2  +  6a  -  7  and  a2  +  8a  +  7. 

(Sections   XI-XIV.) 

39.  Eeduce  -t^-tt-t  to  its  lowest  terms. 

,  „    „  ,         2afc*     d*  —  x2     xn~  +  4x  +  3        ,   ,    , 

40.  Eeduce  — -,,  — •,  — — ^  — o  each  to  lowest 

15a2 '    a  +  2;  '  ar  +  oz  +  2 

terms. 

41.  Add  together  — ,   -j- ,    — . 

&  x      4x     5x 


42.  Add  together 


a  —  b'   a  —  b  '   a  —  b' 


c  c 

43.  Add  together and  — ■ — . 

&  a  —  x  a  +  x 

44.  Add  together  ? ,    — ™  >    a 

6  a  +  0     a  —  0      a  —  0 


4o.  Add  T  .   T,  and  -, r 

x  +  1     x  —  1  a;2  —  1 


,„    -r,          #  +  8       ,,      ,  #  —  7 
46.  From r  subtract  -  — s 

a:  —  2  #  —  2 


120  MISCELLANEOUS  EXAMPLES. 


47.  From  • take 


x  —  1  x  +  1' 

JO    -,         a  +  1  ,  ,  ft  —  1 

48.  From take  T. 

a  —  1  «  +  1 

7ft  -  4       lift  -  7 


49.  Simplify 


5 


50.  Simplify   1  -  -^-  and   1  +  ~- 
1     J  1  +  ft  1  —ft 


(Sections    XV-XVII.) 


51.  Multiply  __  +  _Ibys. 

52.  Multiply  2a~  X  by  17. 

ro  tvi-  14.-  i  ^2  +  2^  +  1  .     ,-r  —3x  +  2 

53.  Multiply  ^_5^+6  by^  +  4^  +  3 

K.  r„  ft2        62        c2 

54.  bimplily  -7-  x  —  x  — r  . 

r     J  oc       ac       ab 

KK  D.       ,.„  ft  +  1       ft  +  2        ft  —  1 

55.  Simplify        — -  x  — — r-  x 


ft  -  1      ft2-  1       (ft  +  2)2 


K_  ...     4ft2^   .      %a¥ 

56-  Dmde6^byi5^- 

57.  Divide  -= 7-  by  -  — T. 

a~  —  o~     J  ft  —  o 

58.  Divide  -^-  by  — —  -^ . 

2^2        2 


MISCELLANEOUS  EXAMPLES.  121 

59.  Divide  — ^  by  1 ^r. 

a  +  1     J  a  +  1 

GO.  Divide  a  +  1   by  ^±1  +  1. 

_.     _.  .,     a'2  —  4a  +  4    ,      a'  —  3a  -f  2 

61.  Divide  -^ _ —  by  —, -. — . 

aJ  -  6x  +  0      J  xl  —  4a  +  3 

62.  Find  the  numerical  value  of  3a  —  25  -f  2c  —  (45  — 
(5c  —  M))   when  a  =  4,  5=1,  c  =  —  1,  d  =  0. 


He 
2c -35 


63.  Find  the  value  of   — — l7tt   when   5  =  3,    c  =  7. 


,    -r,.    -,  xi  •    i      !        .  2«  +  2      %a  -  9    . 

64.  Find  the  numerical  value  of ^-  H ~  when 

a  —  3         rt  —  2 

a  =  4. 

65.  Find  the  numerical  value  of  3a  —  £(35  —  7(c  —  J)) 
jvhen  «  =  15,  5  =  2,  c  =  3,  d  =  5. 

M    _.    •.  ,.  .     ,      ,  .  «2  +  52      c2-rf    , 

66.  Find  the  numerical  value  of ■—  -\ -n —  when 

c  a 

a  —  \,  5  =  2,  c  =  3,  d  =  4. 

67.  Find  the  numerical  value   of    —  +  -^ when 

a        o        a 

a ■=  1,  5  =  4,  c  =  6. 

68.  Find   the   numerical   value   of   6«5a  +  10^25  —  5c2 
when  a  =  1,  5  =  9,  c  =  8. 

69.  What   is   the   difference   between   4«  and  «4  when 
a  =  2? 

70.  What   is   the  difference  between  a5  and  8a2  when 

«=  8? 

6 


122  MISCELLANEOUS  EXAMPLES. 

(Section    XVIII.) 

Solve  the  equations 

71.  12(3  -  3)  -  3(23  -  1)  +  5x  =  22. 

72.  ~  +  12  =  ^  +  7. 


73. 

^x  —  \x  =  5 1  —  \x. 

74. 

0(3  +  5)  =  8(57-3). 

75. 

104-  10(2^  -1)  =  54. 

76. 

3         5       1 

7x  ~  14  ~"  2  ' 

7^  +  4        83-2        K/ 

77.        2        -  — y—  =  5(40  -  23). 


^43-6       83  -  10 

78. 


79. 


80. 


33  —  4       63  ■ 
60  30 


3+2  3—1 

32  54 


33  —  4       53  —  6 


81.  |(53  -  1)  -  6(22  -  33)  =  2x  -  3. 

Q_     343  -  56        73-3       7ft  -5    ,  OI 

82.  -3^-  -T___r--+2^ 


/*3  3  —  O  J    ,_.  4  < 


MISCELLANEOUS   EXAMPLES. 


z+_l        2(x  +  2)  _  9(s  -  3) 
84#       2       +         3  4       ' 


85>   -3~  +  2-12 3~" 


123 


86. 


5x-  6       2x  -  13        x  +  7 


(Sections    XXII,    XXIV.) 
Find  #  and  ?/  in  the  following  simultaneous  equations 


87.   3x  -  4?/  =  25   ) 
5x  —  2y  —  7     ) 


88. 

X  4-  *?/     -  8 

2  +3    ~8 

«    y  _i 

4       12  ~ 

> . 

89. 

22:  —  3  =  y      J 

90. 

4:x  +  9//  =  12  j 
6a;  -  3y  =  7    )  * 

91. 

82:  -  ty  =  12 

#    —  2;/      2a 
4 

^ 

124 


MISCELLANEOUS   EXAMPLES* 


92. 

5       7           1 

i-M 

93. 

iri=1' 

2        6           J 

94. 

a;  +  |  =  19 

Find  x,  y,  and  2 : 


95.  x  +  y  =  z 

x  —  y  4-  z  =  4 
5z  +  #  +  2  =  20 


(Sections   XX,    XXI,    XXIII.) 

96.  Eight  times  a  certain  number  added  to  16  is  equal 
to  16  times  a  number  one  less.     Find  it. 

97.  Find  a  number  such  that  45  times  the  number  in- 
creased by  60  is  equal  to  500  diminished  by  10  times  the 
number. 

98.  What  two  consecutive  numbers  are  such  that  J  of 
the  larger  added  to  -J  of  the  smaller  is  equal  to  9  ? 


MISCELLANEOUS  EXAMPLES.  125 

99.  If  j\  of  the  larger  of  two  consecutive  numbers 
taken  from  \  of  the  smaller  leaves  7,  what  are  they  ? 

100.  Two  persons  have  equal  sums  of  money,  but  the 
first  owes  the  second  60  dollars ;  when  he  has  paid  his 
debt  the  second  has  twice  as  much  as  the  first.  How 
much  had  each  ? 

101.  Divide  21  into  two  such  parrs  that  one  of  them 
shall  contain  the  other  21  times  exactly. 

102.  Divide  the  decimal  fraction  .07  into  two  other 
decimal  fractions  which  differ  from  each  other  by  .007. 

103.  Find  the  number  which  increased  by  5  is  contained 
the  same  number  of  times  in  45  as  the  same  number  dimin- 
ished by  5  is  contained  in  12. 

104.  A  purse  of  eagles  is  divided  among  three  persons, 
the  first  receiving  half  of  them  and  one  more,  the  second 
half  of  the  remainder  and  one  more,  and  the  third  6. 
Find  the  number  of  eagles  the  purse  contained. 

105.  A  person  possesses  $5,000  of  stock.  Some  yields  3 
per  cent.,  four  times  as  much  yields  3^  per  cent.,  and  the 
rest  4  per  cent.  Find  the  amount  of  each  kind  of  stock 
when  his  income  is  $170. 

10G.  Divide  a  yard  into  two  parts  such  that  half  of  one 
part  added  to  22  inches  may  be  double  the  other  part. 

107.  Divide  $120  among  three  persons  so  that  the  first 
may  have  three  times  as  much  as  the  second,  and  the  third 
one-fourth  as  much  as  the  first  and  second  together. 

108.  Two  coaches  start  at  the  same  time  from  two  places, 


126  MISCELLANEOUS  EXAMPLES. 

A  and  B,  150  miles  apart,  one  travelling  5  miles  an  hour, 
the  other  6J  miles  an  hour.  Where  will  they  meet,  and 
at  what  time  ? 

109.  A  number  is  written  with  two  digits  whose  differ- 
ence is  7,  and  if  the  digits  be  reversed  the  number  so 
formed  will  be  f  of  the  former.  Find  the  original  num- 
ber. 

110.  Divide  200  into  two  parts  so  that  one  of  them  shall 
be  two-thirds  of  the  other. 

111.  A  is  three  times  as  old  as  B  ;  twelve  years  ago  he 
was  eleven  times  as  old.     What  are  their  ages  ? 

112.  A  father  has  five  sons,  each  of  whom  is  three  years 
older  than  his  next  younger  brother,  and  the  oldest  is  four 
times  as  old  as  the  youngest.     Find  their  respective  ages. 

113.  Divide  60  into  two  parts  so  that  the  difference  of 
their  squares  shall  be  1,200. 

114.  Divide  30  into  three  parts  so  that  the  ratio  of  the 
first  two  shall  be  1  :  2,  and  that  of  the  last  two  5  :  3. 

115.  If  Gx  -  3  :  Ax  -  5  :  :  3x  +  5  :  2x  +  3,  find  x. 

116.  Eight  horses  and  five  cows  consume  a  stack  of  hay 
in  10  days,  and  three  horses  can  eat  it  alone  in  40  days. 
In  how  many  days  will  one  cow  be  able  to  cat  it  ? 

117.  One-fourth  of  a  ship  belongs  to  A  and  one-fifth  to 
B,  and  A's  part  is  worth  $6,000  more  than  B's.  What  is 
the  value  of  the  ship  ? 

118.  A  certain  fraction  becomes  -J-  if  2  be  added  to  its 


MISCELLANEOUS  EXAMPLES.  127 

numerator,  and  if  2  be  added  to  its  denominator  it  be- 
comes £.     What  is  the  fraction  ? 

119.  If  a  certain  number  be  multiplied  by  7f,  the  pro- 
duct is  as  much  greater  than  16  as  the  product  of  its 
multiplication  by  2f  is  less  than  110.   What  is  the  number  ? 

120.  The  highest  pyramid  in  Egypt  is  25  feet  higher 
than  the  steeple  of  St.  Stephen's  Church  in  Vienna,  and 
the  height  of  this  last  is  -j-J  of  the  height  of  the  pyramid. 
How  high  is  each  ? 

121.  "  The  clock  has  struck ,"  called  out  the  night- 
watchman.  "  What  hour  did  it  strike  ?  "  asked  a  passer- 
by. The  watchman  replied:  "The  half,  the  third,  and 
the  fourth  of  the  hour  struck  is  one  greater  than  the  hour." 
What  hour  did  it  strike  ? 

122.  Of  a  swarm  of  bees  the  fifth  part  lighted  on  a 
blooming  Cadamba,  and  the  third  part  on  the  blossoms  of 
Silind'hri,  three  times  as  many  as  the  difference  between 
the  first  two  numbers  flew  to  the  flower  Cutaja,  and  the 
one  bee  remaining  hovered  in  the  air  unable  to  choose  be- 
tween the  aromatic  fragrance  of  the  Jasmin  and  the  Pan- 
danus.  "  Tell  me,  beautiful  girl,"  said  the  Brahmin,  "  the 
number  of  bees. " 

123.  A  father  died,  and  left  to  his  two  sons  and  his  wife 
$30,000,  with  the  conditions  that  the  share  of  the  elder 
brother  should  be  to  the  share  of  the  younger  as  4:3; 
and  the  share  of  the  mother  should  be  |  of  the  amount 
left  to  both  brothers.     What  was  the  share  of  each  ? 


128  MISCELLANEOUS  EXAMPLES. 

124.  A  farmer  grazes  a  certain  number  of  sheep  and 
oxen  in  two  fields.  One  contains  13  animals,  but  only  half 
the  sheep  and  one-fourth  the  oxen  are  in  it.  The  other 
field  contains  21  animals.  How  many  of  each  sort  had  he 
in  the  fields  ? 

125.  Find  two  numbers  in  the  ratio  of  3  to  4,  whose  sum 
is  to  1  as  30  added  to  the  second  is  to  2. 

120.  If  8  times  one  number  be  taken  from  7  times 
another  number  3  remains  ;  and  9  times  the  first  added  to 
6  times  the  second  is  96.     What  are  the  numbers  ? 

127.  A  said  to  B:  "  If  you  give  me  7  dollars  of  your 
money,  then  I  shall  have  twice  as  much  as  will  remain  to 
you."  B  said  to  A  :  "If  you  give  me  4  dollars,  then  I  shall 
have  twice  as  much  as  remains  to  you."  How  much  had 
A  and  B,  each  ? 

128.  The  sum  of  two  numbers  is  40,  and  their  quotient 
is  3.     What  are  the  numbers  ? 

129.  Find  two  consecutive  numbers  whose  product 
diminished  by  20  equals  the  square  of  the  first. 

130.  The  hour  and  minute  hands  of  a  clock  are  together 
at  12  o'clock,  when  will  they  be  together  again  ? 

If  x  =  number  minute-spaces  gone  over  by  minute-hand  before 
they  are  together  again,  then,  ^x  —  number  minute-spaces  gone 
over  by  hour-hand  before  they  are  together  again.  And  \lx  =  gain 
of  minute-hand.     And,  since  this  gain  must  be  60  minute-spaces, 

.-.    \\x  =  GO. 


MISCELLANEOUS  EXAMPLES.  129 

(Sections    XXV,  XXVI.) 
Find  the  squares  of  the  following  quantities  : 

131.  (1),  x*  -  3x  +  4.     (2),  4a  -  2b  +  3c:    (3),  x  - 

a  +  2b  —  c. 

Find  the  square  roots  of  the  following  quantities  : 

132.  49za  +  12Qax  +  81a2. 

133.  121a2  -  3S0ab  +  22562. 

134.  400«V  -  200abx  +  25b\ 

135.  4a2  -  12ab  +  W  +  20ac  -  SObc  +  2bc\ 

136.  x*  --  Ax3  +  Gx"  -  \x  +  1. 

(Sections  XXVII-XXIX.) 
Find  the  value  of  x  in  each  of  the  following  equations  : 

137.  {x  +  3)2  =  C>x  +  25, 


138.  ~—  +  ---  =  8. 
1  -}  -a;      1  —  x 


139.  i(9  -  2x2)  =  |  -  TV(7z2  -  18). 


-  .A    2  -  ar        3  -  x*        4  -  or       or  -  5        3 
140.   -—  +   -4—  +  -g-  =  _,-  -  T. 


141.  2.t2  -  12a;  +  16  =  160. 
6* 


]30  MISCELLANEOUS  EXAMPLES. 

142.  4z2  -  32^  +  40  =  76. 

143.  x*  -  x  -  40  =  170. 

144.  3a;2  +  2x  -  9  =  76. 

145.  x2  -  %bx  =  a"  -  h\ 


146.   16  -  %■  =  ^  +  7|. 


147.  a;'  -  -f-  =  14a  +  10. 

o 

148.  There  are  three  numbers  in  ratio  J  :  |  :  J,  the  sum 
of  whose  squares  is  724.     Find  them. 

149.  Find  the  number  which  added  to  its  square  gives 
182. 

150.  There  are  two  numbers  one  of  which  is  f  of  the 
other,  and  the  difference  of  their  squares  is  63.  Find 
them. 

151.  There  is  a  rectangular  bathing  pool  whose  length 
exceeds  its  breadth  by  ten  feet,  and  it  contains  1,200  square 
feet.     Find  its  length  and  breadth. 

152.  The  difference  of  two  numbers  is  5,  and  their  prod- 
uct is  1,800.     Find  them. 

153.  The  product  of  two  numbers  is  126,  and  if  one  be 
increased  by  2  and  the  other  by  1,  their  product  is  160. 
Find  them. 

154.  Find  two  consecutive  numbers  whose  product  is 
600. 


MISCELLANEOUS  EXAMPLES.  131 

(Sections    XXX,    XXXI.) 

Find  x  and  y  in  the  following  simultaneous  equations  : 

155.  x  —  y  —  15  ) 

156.  z  +  y  :x  —  y  :  :  13  :  o  ) 

y-  +  x  =  25  J 

157.  y"  -  lOz  =  lOy  +  36  ) 

a;  +  2y  =  3o  J  • 

158.  3x  +  2y  =  20  ) 

zx*  -  f  =  n  \  ' 

159.  Beduce  a/45  -  V%0  +  a/50  +  yl25  -  V180. 

160.  Square  1  +  a/3  . 

161.  Multiply  4  -  V3   by  4  +  i/3. 


162.  Simplify  a/512oW 

163.  Simplify  yf  • 

164.  Simplify 


165.  Simplify 


3- V5 
4 


a/5-1 
166.  Multiply  a/3  +  2a/2~  by  2a/3~  +  V% 

Find  a?  in  the  following  equations  : 


167.   Vx  +  Vz  -  7  =  7. 


132  MISCELLANEOUS  EXAMPLES, 


168.    a/5z  +  11  +  a/&»  —  9  =  10. 

, .  , 2 

1G9.    ^x  +  1  +  Vv  -  1  =  -7==- 

V#  +  1 


(Section    XXXII.) 

170.  Compare  the  ratios 

4  :  5  and  15  :  1G  ; 
14  :  15  and  22  :  23. 

171.  What  is  the  ratio  of  a  inches  to  c  feet  ? 

172.  Find  a  fourth  proportional  to  f,  f ,  \. 

173.  What  number  is  that  to  which  if  2,  4,  and  7  be 
severally  added,  the  first  sum  is  to  the  second  sum  as  the 
second  is  to  the  third  ? 

174.  What  two  numbers  are  those  whose  difference, 
sum,  and  product  are  proportionate  to  the  numbers  3,  5, 
and  20,  respectively  ? 

(Sections    XXXIII,   XXXIV.) 

175.  Find  the  sum  of  the  progression  1,  7,  13,  19,  etc., 
to  50  terms. 

176.  Find  the  sum  of  f,  if,  1|,  etc.,  to  20  terms. 

177.  Insert  five  arithmetical  means  between  12  and  20. 

178.  The  first  and  last  of  30  numbers  in  arithmetical 
progression  is  2^  and  2J.  What  are  the  intervening 
terms  ? 


MISCELLANEOUS  EXAMPLES.  133 

179.  A  certain  number  consists  of  three  digits,  which 
are  in  arithmetical  progression  ;  and  the  quotient  of  the 
number  divided  by  the  sum  of  its  digits  is  15  ;  but  if  39G 
be  added  to  it,  the  digits  will  be  inverted.  What  is  the 
number  ? 

Let  x  —  the  middle  digit,  and  y  —  the  common  difference  ;  then 
x  —  y,  x  and  x  +  y  =  the  digits,  respectively  ;  and  the  number  = 
100  (x  -  y)  +  10s  +  (x  +  y)  =  111  x  -  99y. 

Hence,  the  simultaneous  equations  : 

Ilia;  -  mj_ 
3x 

Ilia;  -  ddy  +  896  =  100  (x  +  y)  +  10z  +  (x  -  y). 

180.  The  population  of  a  town  increases  in  the  ratio  of 
-jV  annually;  it  is  now  10,000.  What  will  it  be  at  the  end 
of  four  years  ? 

181.  Find  the   geometric    mean   between    - —        and 


182.  Insert  two  geometric  means  between  5  and  —  |. 

183.  Insert  3  geometric  means  between  12  and  972. 

184.  Find  the  value  of  the  recurring  decimal  .8181.8181 
as  a  decreasing  geometrical  progression. 

185.  A  farmer  sowed  a  bushel  of  wheat  and  used  the 
whole  produce,  15  bushels,  for  seed  the  second  year,  the 
produce  of  this  second  year  for  seed  the  third  year,  and 
the  produce  of  this  again  for  the  fourth  year.  Supposing 
the  increase  to  have  been  always  in  the  same  proportion 
to  the  seed  sown,  how  many  bushels  of  wheat  did  he 
harvest  at  the  end  of  the  fourth  year  ? 


134  GENERAL  REVIEW  QUESTIONS. 

SECTION    XXXVI. 
GENERAL  REVIEW  QUESTIONS. 

I. 

1.  How  are  quantities  represented  in  algebra  ?      Why 
do  we  use  letters  ? 

2.  What  is  the  chief  point  of  distinction  between  digits 
and  letters  as  used  in  algebra  ? 

3.  What  are  the  signs  of  addition  and  subtraction  ? 

4.  What  are  positive  quantities  ?   Negative  quantities  ? 

5.  What  are  the  signs  of  multiplication  ? 

6.  When  may  we   omit  them  and  still  denote  multipli- 
cation ? 

7.  What  is  the  difference  between  345  and  abc  when 
a  =  3,  b  =  4,  c  .-=  5  ? 

8.  What  are  the  signs  of  division  ? 

II. 

1.  AVhat  is  a  factor  ?     What  is  a  coefficient  ? 

2.  What  is  a  power  ? 

3.  What  is  an  index  ?    What  other  name  have  we  for 
the  index  of  a  power  ? 

4.  What  is  an  algebraic  expression  ? 

5.  What  are  the  terms  of  an  expression  ? 


GENERAL  REVIEW  QUESTIONS.  135 

6.  What  is  a  monomial  ?  a  binomial  ?  a   trinomial  ?  a 
potynomial  ?     Give  an  example  of  each. 

7.  What  are  like  terms  ?     Unlike  terms  ? 

8.  How  do  we  simplify  an  algebraic  expression  ? 

9.  What  is  the  rule  for  addition  ? 

10.  What  is  the  rule  for  subtraction  ? 

11.  Show  the  reason  for  the  rule  of  the  signs  in  sub- 
traction. 

12.  How   does    algebraic    addition   differ   from   arith- 
metical ?     Illustrate  by  an  example. 

13.  How  does   algebraic  subtraction  differ  from  arith- 
metical ?     Illustrate  by  an  example. 

III. 

1.  State  and  prove  the  rule  of  the  signs  in  multiplica- 
tion. 

2.  How  do  we  multiply  monomials  together  ? 

3.  Give  the  rule  for  multiplying  a   polynomial  by  a 
monomial. 

4.  Give  the  rule  for  multiplying  a  polynomial  by  a 
polynomial. 

5.  State  and  prove  the  rule  of  signs  in  division. 

G.  What  is  the  rule  for  dividing  a  monomial  by  a  mo- 
nomial ?     A  polynomial  by  a  monomial  ? 

7.  Give  the  rule  for  the  division  of  polynomials. 


136  GENERAL  REVIEW  QUESTIONS. 

8.  What  is  meant  by  arranging  a  polynomial  with  ref- 
erence to  a  certain  letter  ?     Give  an  example. 

IV. 

1.  What  is  the  square  of  the  sum  of  two  quantities 
equal  to  ? 

2.  Express  this  by  a  formula,  when  a  and  b  are  the 

quantities. 

3.  What  is  the  square  of  the  difference  of  two  quantities 
equal  to  ? 

4.  Express  this  by  a  formula. 

5.  What  is  the  product  of  the  sum  and  difference  of 
two  quantities  equal  to  ? 

6.  Express  this  by  a  formula. 

7.  What  are  brackets  ? 

8.  Give  the  rules  for  removing  brackets. 

9.  Show  by  examples  how  to  remove  brackets  with  the 
plus  sign  before  them  ;  with  the  minus  sign  before  them  ; 
with  the  sign  of  multiplication  before  them. 

V. 

1.  What  is  the  product  of  x  +  a  by  x  +  h  ?     Of  x  —  a 

by  x  —  b  ?     Of  x  +  a  by  x  —  b  ? 

2.  What  are  the  factors  of  x"  +  5x  +  G  ?  Of  x1  ~  13a  + 
40  ?     Of  a?  +  bx  -  6  ?     Of  x1  -  x  -  30  ? 

3.  What  are  the  factors  of  x2  +  4z  +  4  ?  Of  x*  -  6x  + 
9  ?    Of  4z2  -  9aa  ? 


GENERAL  REVIEW  QUESTIONS.  137 

4.  What  is  the  least  common  multiple  of  two  algebraic 
expressions  ? 

5.  Give  the  rule  for  finding  this  L.  C.  M. 

6.  What  is  the  G.  0.  D.  of  two  algebraic  expressions. 

7.  Give  the  rule  for  finding  it. 


VI. 

1.  Do  the  rules  for  the  operations  of  reduction,  addi- 
tion, multiplication,  etc.,  on  algebraic  fractions  differ 
from  those  in  the  arithmetic  for  common  fractions  ? 

2.  Give  these  rules. 

3.  What  is  meant  by  the  numerical  value  of  an  alge- 
braic expression  ?  and  how  do  we  find  it  in  any  supposed 
case  ? 

4.  What  is  the  difference  between  a%  and  2#2,  when 
a  =  2? 

5.  What  is  the  value  of  <r  —  2ab  +  &2,  when  a  =  G 
and  b  —  5  ? 

VII. 

1.  What  is  an  equation  ? 

2.  What  is  an  identity  ? 

3.  What  is  a  known,  and  what  an  unknown  quantity  ? 

4.  What  is  a  simple  equation  ? 

5.  What  is  meant  by  the  solution  of  an  equation  ? 

6.  Enumerate  the  operations  which  may  be  performed 


138  GENERAL  REVIEW  QUESTIONS. 

on  an  equation  without  destroying  the  equality  of  the  two 
sides. 

7.  Give  the  rule  for  the  solution  of  a  simple  equation 
with  one  unknown  quantity. 

8.  When  is  the  solution  of  an  equation  said  to  be  veri- 
fied. 

9.  Give  the  rule  for  solving  problems  by  equations. 

10.   Give  some  examples  of  translation  from  common 
into  algebraic  language. 

VIII. 

1.  What  are  simultaneous  equations  ? 

2.  What  are  the  different  methods  of  eliminating  one 
of  the  unknowns  in  simultaneous  equations  ? 

3.  Explain  these  methods. 

4.  How  do  we  proceed  when  we  have  three  simulta- 
neous equations  with  three  unknown  quantities  ? 

IX. 

1.  What  is  meant  by  involution  ? 

2.  What  is  the  rule  for  the  square  of  the  sum  of  three 
quantities  ?     Give  the  square  of  a  +  b  +  c. 

3.  What  is  meant  by  evolution  ? 

4.  What  is  the  square  root  of  a  number  ?     The  cube 
root  ?     Fourth  root  ? 

5.  What  is  the  Radical  Sign?    How  do  we  indicate 


GENERAL  REVIEW  QUESTIONS.  139 

the  square  root,  cube  root,  fourth  root,  etc.,  of  a  quan- 
tity? 

6.  What  sign  has  the  square  root  of  a  quantity  ?  The 
fourth  root  ? 

7.  Why  has  a  negative  quantity  no  square  root  ? 

X. 

1.  How  do  we  find  the  square  root  of  a  monomial  ?  of 
a  fraction  ? 

2.  When  is  a  trinomial  a  perfect  or  complete  square  ? 

3.  How  do  we  find  the  square  root  of  a  trinomial, 
when  it  is  a  perfect  square  ? 

4.  What  term  added  will  render  a  binomial  of  the  form 
x2  +  px,  or  x~  —  px,  a  perfect  square?  Illustrate  by  an 
example. 

5.  Give  the  rule  for  finding  the  square  root  of  a  poly- 
nomial. 

XL 

1.  What  is  a  quadratic  equation,  or  equation  of  the 
second  degree  ? 

2.  What  is  a  pure  quadratic  ? 

3.  What  is  an  affected  quadratic  ? 

4.  How  many  values  has  the  unknown  in  a  quadratic 
equation  ? 

5.  How  do  we  solve  a  pure  quadratic  ? 


140  GENERAL  REVIEW  QUESTIONS. 

6.  Give  the  steps  in  the  solution  of  an  affected  qua- 
dratic, and  the  reasons  for  them. 

7.  In  the  solution  of  simultaneous  equations  with  two 
unknowns  by  quadratics,  what  is  the  usual  mode  of  elimi- 
nation ? 

XII. 

1.  What  is  a  radical  expression,  or  surd  ? 

2.  What  is  a  radical  of  the  second  degree  ? 

3.  What  is  the  coefficient  of  a  radical  ? 

4.  What  is  the  difference  between  2a/^  and  2  +  \/x 
when  x  =  81  ? 

5.  What  is  the  difference  between  \/a  +  b  and  V#  + 
b,  when  a  =  9  and  b  =  16  ? 

6.  What  is  the  difference  between  \  —  and  —r-  when 

b  o 

a  =  36  and  b  =  9  ? 

7.  How  may  we  transfer  the  coefficient  of  a  radical  of 
the  second  degree  as  a  factor  under  the  radical  sign  with- 
out affecting  the  value  of  the  expression  ?  Give  an  ex- 
ample. 

8.  How  may  we  transfer  a  square  factor  from  under 
a  radical  sign  of  the  second  degree  as  a  coefficient  before 
it  ?     Give  an  example. 

9.  How  do  we  use  this  to  simplify  a  radical  of  the 
second  degree  ? 

10.   What  are  similar  radicals   of   the  second  degree  ? 
How  do  we  add  or  subtract  them  ? 


GENERAL  REVIEW  QUESTIONS.  141 

11.  How  do  we  multiply  two  radicals  of  the  second  de- 
gree ?     How  do  we  divide  them  ? 

12.  How  do  we  simplify  radical  expressions  of  the  form 
y  —  ?     Illustrate  by  a  numerical  example. 

13.  How   do    we    simplify    expressions    of    the   form 

c  c 

~7P ,  or  j=  ?    Illustrate  bv  an  example. 

a  +  yb  a  —  yb  "  L 


14.  How  do  we  solve  equations  of  the  forms  \/x  +  b 


c,  yx  +  a  +  ^/x  +  b  —  c  ? 

XIII. 

1.  What  is  ratio  ? 

2.  What  is  proportion  ? 

3.  What  three  important  things  are  to  be  remembered 
in  operating  on  ratios  and  proportions  ?     (See  Art.  117.) 

4.  How  do  we  simplify  a  ratio  ? 

5.  What  are  the  extremes  and  what  the  means  of  a  pro- 
portion ? 

6.  What  equation  holds  between  them  ? 

7.  When  are  three  quantities  said  to  be  in  proportion  ? 
How  do  we  find  the  mean  proportional  between  two  quan- 
tities ? 

8.  What  is  the  single  Rule  of  Three  ? 

9.  Give  the  leading  proportions  which  may  be  obtained 
from  the  proportion  a  :  b  :  :  c  :  d. 


142  GENERAL  REVIEW  QUESTIONS. 

XIV. 

1.  What  is  an  arithmetical  progression  ?  Increasing  ? 
Decreasing  ?     In  what  general  forms  may  it  be  written  ? 

2.  How  do  we  find  any  term  of  an  arithmetical  pro- 
gression when  the  first  term  and  common  difference  are 
given  ? 

3.  Give  the  rule  for  finding  the  sum  of  the  terms  of  an 
arithmetical  progression. 

4.  How  do  we  insert  a  given  number  of  arithmetical 
means  between  two  numbers  ? 

5.  How  do  we  find  the  arithmetical  mean  or  average  of 
two  numbers  ? 

XV. 

1.  What  is  a  geometrical  progression  ?  Increasing  ? 
Decreasing  ?     In  what  general  form  may  it  be  written  ? 

2.  How  do  we  find  any  term  of  a  geometrical  progres- 
sion when  the  first  term  and  ratio  are  given  ? 

3.  Give  the  rule  for  finding  the  sum  of  the  terms. 

4.  Give  the  rule  for  finding  the  limit  of  the  sum  of  the 
terms  of  a  decreasing  geometrical  progression  when  the 
number  of  terms  is  infinite. 

5.  How  may  this  rule  be  applied  to  find  the  limiting 
value  of  a  circulating  decimal  ?    Illustrate  by  an  example. 

0.  How  do  we  find  the  geometric  mean  between  two 
numbers  ? 

7.  How  do  we  insert  two  geometric  means  between  two 
numbers  ?     How  do  we  insert  three  ? 


ANSWERS.  143 


ANSWERS. 

Examples — 2,  page  12. 

1.  12a.      2.  2x  +  2y  +  2*.      3.  2  +  5«.     4.  2«3  +  263. 
5.    3a;4  -  x2  -  15*  -  2.         6.    W  +  2a2fl  +  dab2  -  b\ 

7.  19«*/2. 

Examples — 4,  page  14. 

1.  b  +  %.      2.   66  -  2c.      3.  7a  -  be.      I.  a  -  x  -  96. 
5.     4a-2.       6.    2m?*  +  4ra  -  7*fc       7.     tort  4-  We  +  2c2. 

8.  -J«6  —  be  +  2.       9.  a3  —  a  +  5a2a;  +  lllax2  +  166a;3. 

Examples — 5,  page  15. 

2.  36a2  -  63ab.      3.    -  60ab  +  96a2.     4.  40a;4  -  20ax3 

—  12aV.  5.  -  4abx  +  8aca;  -  12bdx.  6.  3a9c  +  6afo. 
7.  -  4a&Y  -  106a:?/3  +  Gcxy2z.  8.  -  21a;10  +  14a;7  - 
28a;4. 

Examples— 6,  page  17. 

1.  ac  +  ay  +  ex  +  ar?/.      2.  5a;2  —  6a;  —  8.      3.  x2  —  x 

—  20.  4..  6x2  -  17a;  +  12.  5.  x  -  3x2  +  2a;3.  6.  ac3  - 
tfbc  _  JV  +  ab\      7.   110a'2  +  118«a;  +  24^2.      8.  x  +  3a;2 

—  xy  —  y  —  2if.  9.  2<r6  —  3«6c  +  2«2c  —  «62  +  &ac. 
10.  a;4  -  1.      11.  25  +  6x2  +  a;4.      12.  a;2  +  Sx  +  16  -  /. 


*44  ANSWERS. 

13.  9aV  -  Wy\  14.  2a;4  -  32.  15.  abxb  +  y\  16. 
a4  +  4J\  17.  a?  +  63  +  c3  -  3aJc.  18.  1  +  6a°  + 
5a6.     19.  a;4  -  10a;3  +  25a;2  -  81.     20.  a?"  -  6a?a  +  12a;  -8. 

Examples — 7,  page  18. 

1.  3.     2.  -  2k     3.   -  6?/.      4.  £3.     5.  5.     7.  8a  -  5b. 

Examples— 8,  page  20. 

1.  4a;  -  32.      2.  0.     3.  125  -  5a;2.      4.  2cx.      5.  -  4. 

Examples — 9,  page  23. 

In  these  examples  the  pupils  should  give  the  answers 
orally. 

Examples — 11,  page  25. 

1.  2b  +  3c  —  4o\    2.  —  a-bx  +  cy.     3.  —  3aa;  +  46  — 

a;2.     4.  1  +  7ac  -  2bc. 

Examples— 12,  page  26. 
1.  x  +  2.  2.  a  +  1.  3.  3a  +  a;.  4.  a?  —  8.  5.  a;  + 
#  +  z.  ft.  a  +  b  +  c.  7.  «  —  6.  8.  a2  —  ab  +  62. 
9.  a  -  b.  10.  a;2  +  x  +  3.  11.  x  —  y-  z.  12.  5a2  + 
3.r2.  13.  a  -  b  -  c.  14.  3a;2  -  x  +  2.  15.  3a;2  -  2abx 
-  2cfb\  18.  dab  -  ix.  19.  4ar  +  Sax  +  9a2.  20.  8# 
+  12a2  +  18a  +  27. 


ANSWERS.  145 

Examples — 13,  page  29. 
These  answers  the  pupil  should  give  orally. 

Examples — 14,  page  31. 

1.  16a*b.     2.  2Cmx\     3.   12<//2.      4.  x  +  1.      5.  x  -  2. 
C.  x  +  9.     7.  x  -  1.     8.  dx  -  4.     9.  x  -  4. 

Examples — 15,  page  33. 

1.  Uab.     2.  2a2b2c.     3.  240ft4.      4.  cfbc2.      5.  12Qx4y*. 

6.  126ft6.     7.  ZihtW.     8.  ft(:t2  -  if).   9.  6(ft2-  62).   10.  x* 
-  4x-2  +  6x  -  2. 


Examples — 16,  page  34. 

f     ft      a            a  —  b    a  —  b      „    5J9  1        .    #  +  # 

6>         ft                         />                   fl                        OftC  2UC               X— if 

4ft  —  bb        „    x  —  y    x  —  1       _     x  —  a  x  —  2 

20c       '        *        #      '  2:  —  2  '           a:  +  ft '  £  +  2 ' 


Examples — 17,  page  36. 
j    5x     Sx      x  9     12ab*x       4bc  -f  kac      3a*x 


15'    15'    15*  48c£2  '        48c^      '    48c.r 

3    ^-^         2ft2  _6_      _3_      _2_  10a  +  8 

a  —  x  '    a  —  x'  6x '    (Jx '    Qx  '  18 

10a;  +  17        a    65ftz  -  13ft        2x  +  4 
18      *       b*  "     52ft2        '        52ft2    ' 


14G  ANSWERS. 

Examples — 18,  page  38. 

i     %  +V  +  *       «    3«  +5       „    7a?  .     13             5a: 

I.      — .       Z.     p: .        O.     -pr-  .  4.     r—v        .       0.    -77- 

a                       3a?                  9  12a             3 

„    8«Z>  +  7«        .    3a  +£           13a?  -2  Q    24/;  -- 23 


246c  i  24  0 

10.  »         11.^+1.       12.   *         13.  0.       14.      15 


'  3  '  c  3  "  x- 1 

15.  5 -.  16-  2TV 


Examples — 19,  page  40. 

t      x  x  m    3a?  -f  5  .3  ,    2c 

'•  u-     2-  ft'       3  ~6—       4-  io-       5  y 

—    4f0C  m       r\  c  —   </ClX 

.   -= r¥.        7.  0.        8. 


10a?  4-  15 

a?2- 

-4. 

10  - 

-  2a? 

a'  -b*  x'  +  3.i;  +  2 

10.  -toy~y\  „  _4tf             12 

a?2  +  a?#  a  —  b                             25 

13.gV~a-c.  14.0. 


Examples — 20,  page  42. 

a  +  b  c  2 

9  —  4a?  15aa?  1  1  1 

_1  (^ -2)  (a? -2)        (a?  -  2)  (a?  -  5) 

"'   1-a*'  (a? -3)  (a- -5)       (a?-l)  (a?  -  2)  ' 


„.        (a?  -  2)  (x  -  2) 
CanCGllln^  }*  -  1    (x  -  8 


ANSWERS.  147 


Examples — 21,  page  43. 


11 


I. 

bx 

cy ' 

2. 

ax 

w 

3. 

3x 

28* 

4. 

±         5 

25?wJ 
3 

1 

-3c 

c 

• 
7. 

-3, 

7#2.      8. 

2axby 
be     ' 

i 

}    2x  +  l 
'    x-V 

10.  15. 

wo 

a  —  b 
a 

.     12 

(x  + 

(x 
'   (* 

3)2 

+  1)  (x 
-3)(x 

(X 

-  3)  (x  + 
+  1)  (S  + 

3) 

Examples — 22,   page  45. 

1.-1        2.-3.         3.    -  6.         4.   -  8.  5.   -  5. 

6.   3.       7.   6  ;  24  ;   -  17.       8.   18  ;  -  105.       9.  -  112f 

10.  41.          11.   11;    19.          12.   -If          13.  5;   23f. 

14.  80  ;  160.          15.   -  f  If.          16.  125  ;  8.  17.  3  ; 
9f.        18.  &. 

Examples — 23,  page  49. 

1.  6  ;  -  4.        2.  3  ;  8.        3.   11 ;  6}.        4.  2G.1 ;  8|. 
5.  -1. 

Examples — 24,  page  50. 

2a  +  b  —  c      c 


1.  b  —  a  ;  c  +  a  2. 


2  '    a 


m  —  n  —  p  b  —  a  —  c 

a  -b      '  2~~ 


148  ANSWERS. 

Examples — 25,  page  50. 
1.  5  ;  4.         2.   -  2  ;  6.         3.  2.         4.  5. 

Examples — 26,  page   52. 
1.   15,  GO.  2.  20.  3.  144.  4.   18.  5.   2f. 

6.  — .  7.  51|.  8.  4.  9.  2.  10.  8.  11.  1. 
12.  8,  m.         13.  1.         14.   18. 

Examples — 27,  page  58. 

1.   9.  2.   20.  3.  9,  24.  4.    205,    615,    2460. 

5.  48.  6.  GO.  7.  48.  8.  14,  15,  1G.  9.  17,  18. 
10.  6,  18.  11.  8,  20.  12.  40.  13.  80  years,  20 
years.       14.  18  women,  22  men,  50  children.         15.   $48. 

Examples — 28,  page  61. 

1.  24,  25,  26,  27.  2.  2T2T  hours.  3.  13||.  4.  3 
hours,  9  miles,  and  12  miles.  5.  2-fo  days.  6.  147 
in  each.        7.  2  miles.         8.  f.      9.  336,292.        10.  28. 

Examples — 29,  page  66. 

1.  x  =  6,  y  =  9.  2.  x  =  17,  #  =  7.  3.  z  =  50, 
y  -  20.  4.  a  =  4,  y  =  2.  5.  3  =  4,  ?/  =  5.  6.  x  = 
15J,  y  =  -  -V .       7.  ^  =  9,  y  =  11.       8.  &  =  8,  #  =  12. 


ANSWERS.  149 

9.  x  =  11,  y  =  7.  10.  x  =  15,  y  =  2.  11.  x  =  3, 
?/  =  7.  12.  ^  =  5,  y  =  3.  13.  x  =  1800,  7/  =  100. 
14.  x  =  8,  y  -  5.  15.  a;  =  19,  ?/  =  3.  16.  x  =  7, 
y  =  9.      17.  x  =  50,  y  =  271      18.  x  =  12,  7/  =  4. 

Examples — 30,  page  68. 

1.  24,48.          2.  13.  3.  18,    28.          1.  880,    $40. 

5.  §8,000,  §10,000.  6.  620,7641,   $15,235f.         7.  /0. 

8.  24  cows,  36  horses.  9.  24,  6.  10.  32  in  first  class,  54 
in  second  class. 

Examples — 31,  page  71. 

1.  x  =  2,  y  =  3,  z  =  4.     2.  x  =  5,  y  —  —%,  %  —  _  3. 
3.   a  =  6,     //  =  2.    z  =  4.  4.   2  =  5,    y  =  8|  «  =  9. 

5.  a;  =  6,    ^  =  18,  2  =  10.       6.  x  =  1,  y  =  2,    s  ==  3. 


Examples— 32, 

page  76. 

I.    27tf3Z>\         2.     -8/rW. 

3.     ~.       4.    Ma*b\ 

lbc 

64«4&V.      6.    ^V.         7. 

%  .        8.  a'  +  4a  +  4. 
4a; 

4/,V  +  4§/>  +  1.     10.   9ma  - 

30//?;/    f  25 //'■".       11.   «Vaj" 

+  2ffca#  +  y'.      12.     9      +     3  •  +  ^ \ 


150  ANSWERS. 

Examples — 33,  page  79. 

I.    ±5ay.        2.    ±10a?xy.         3.    ±  7ab\        4.    ±  ~. 

loy 

5.    ±£L.     6.    ±*£.     7.   ±^.     8.  ±f     9.±i. 
7«8y*  2a  4  3 


10.    ±f. 


Examples— 34,  page  80. 


1.    ±  (a  +  1).  2.    ±  («  -  2).  3.     ±  (a;  +  f). 

4.     ±  (3a£  -  &).       5.     ±  (4a  -  3b).       6.     ±  (8a:2  -  §6). 
7.    ±  (a  +  i).       8.    ±  (4*  -  1).      9.  ±  (x  -  |). 

Examples — 35,  page  81. 

1.     ±  {x  +  3).             2.     ±  (a;  -  6).  3.     ±  (x  -  V). 

4.    ±  (s-  i),  5.     ±  (y+  {).  ti.     ±(*-|). 

7.     ±  (a  -  i).  8.     ±  (*  -  2a).  9.    ±  (y  +  f). 

10.    ±  (*-■&).  11.    ±  («-*).  12.    ±  (ar  +  j|. 

Examples — 36,  page  83, 


1.   8«  +  95. 

2.    x1  +  2aa;  +  a\ 

3.  a;2  -  2a;  —  2L 

4.   3«  -  2/y  +  c. 

5.  3«'?  -  2«  +  1. 

6.  4^2  -  2b  +  c2. 

7.  3r  -  «  +  2. 

8.    a  —  b  +  2c. 

9.    as2  —  x  —  1. 

r2 

11 

.  a2  —  5a  +  G. 

ANSWERS.  151 

Examples — 37,  page  85. 

1.  ±VT-  2.  ±V%  3.  ±-V-.  *  ±2.  5.  ±  5. 
0.    ±  6.      7.    ±  |.      8.    ±8.      9.    ±  4.      10.    ±  2. 

Examples — 38,  page  88. 

1.  15  or  3.  2.  4  or  -  1.  3.  5  or  -  1.  4.  -  1  ±  V'f . 
5.  3  ±  V  -4^.  6.  7  or  -  6.  7.  f  or  4.  8.  10  or  -  V. 
9.  TV  or  -  T\.  10.  -  1  or  V5.  11-  It  01*  -  1- 
12.  -  f  or  2.  IS.  -  I  or  +  9.  14.  -  7  or  +  0. 
15.  2  or  -  f.    16.  4  or  %»-.    17.  5  or  3. 

Examples — 39,  page  90. 

1.28.  2.9.  3.9.  4.100.  5.9,15.  i.  \  or  f 
7.  10,  5.      8.  10,  11.      9.  3.      10.  25.      In  these  answers 

the  negative  results  are  omitted. 

Examples — 40,  page  98. 


1.  x  =  2  or  3) 

2.  ar=6) 

3.  x  =  8  or  6) 

y  =  'd  or2f' 

y  =  5) 

y  =  6  or  8) 

4.  x  =  8  or  I        ) 

5.  a?=  ±'l2) 

6.  a-  =  ±  24  j 

y  =  3  or  -  4± ) 

y  =  ±  36  ) 

y  =  ±  SO) 

7.  a;  =  ±  3  £ 

8. 

a?  =  11 ) 

9. 

*  =  4«r-«* 

y  =  ±8)'" 

//  =  8  )  ' 

y  =  5  or   -Vs6-  J 

152 


ANSWERS. 

).  x  =  5  or 

-9    }              11.  x  =  5  or  6T\ 

y  —  2  or 

-  12  ) "                  y  =  2  or  ft 

Examples — 41,  page  98. 

1.  2afc8\/7a.       2.  W%  -  a.      3.  3^3,    2^3,    20  V^ 
4.  2V27  5^27   0a/2T        5.  47  v^         0.   12  -  lOV'C. 


7.  3  +  V3.  8.  yV  -  ft*.  9.  2.  10.  7  -  4^/3: 
11.  !a/2L  12.  1^27  13.  t^-a/I^  14.  3V^+  3. 
15.   V3-1.         16.  -KV5  +  V2).         17.  0. 

Examples — 42,  page  100. 

1.  V  -  2bc  +  c*-a.  2.  144.  3.  36.  4.  5.  5.  25. 
6.  3.      7.  5.       8.  2.       9. -16. 

Examples — 43,  page  102. 

1#  *  5  S"       A   4?  ;    3""        d*   J  ;   4ft'       4*  5ft  ;  3^  ' 

K    ft  +  5     2ft  +  b         .    .  w    36      20m 

5-^r;~^-    •■  !-«;«  +  *     '-s'-ts-- 

8.  17  ;  18. 

Examples — 44,  page  105. 

2.  7.  3.  x  :  y  :  :  5  :  4.  4.  11.  5.  4/>m.  6.  8 
and  12.  8.24,30.  9.11.  10.  4  +  x\  11.2 
or  -  12.         12.  21,  22,  23,  24. 


ANSWUES.  153 

Examples — 45,   page  110. 

1.  (1.)  55,  111.  (2.)  44,  86.  (3.)  131,  27J. 
(4.)  135,  65. 

2.  (1.)  590.  (2.)  T240.  (3.)  23G1.  3.  300. 
4.  881.  5.  1,700  yards.  8.  T4F.  9.  a.  10  10J,  10f. 
11.  Com.  dif.  2.  12.  Com.  dif.  ^  series,  3||,  3||, 
etc.     13.  64331. 

Examples — 46,  page  115. 

1.  (1.)  3.     (2).  2.     (3.)  f.     (4>).l.     (5.)  2.     (6.)  3s. 

2.  25.  3.  18,  54.  4.  J,  |.  5.  The  arithmetical. 
6.  25924.       7.  2  dollars   and  55   cents.       8.   J.        9.  f 

10.  if         11.  24,  48,  96. 

Miscellaneous  Examples,  page  117. 

1.  -  Sa'x  +  Sax"  -  2x\  2.  -  Ga2b  -  4«\  3.  ±a  - 
7c.  4.  4rr  -  W.  5.  9«a  -  b\  6.  ft2  +  2ab  +  b*  -  c1  - 
2cd  -  d\         7.  a*  +  artf  +  &\  8.  25a'  +  1000r^  + 

lOOOCZr,  19G«2  -  30Sab  +  12162.      9.  x.       10.  6rtz  -  4.r. 

11.  27a;3  +  9tfy  +  3.tt/'2  +  ?/3.  12.  a3  -  2cC  +  2a  —  1. 
13.  z2  +  to  +  3«2.  14.  'Sa  -  ±x  4-  £.  15.  4.f3  -  Gcz2  + 
2c2x.  16.  8  +  6<c.  17.  1  -  x\  18.  x.  19.  («  -  J) 
(a  +  J)  (a2  +  V).        20.  (3  -  5)  (x  -  6).        21.   (*  -  6) 


154  ANSWERS. 

(x  +  5).  22.  (2m  +  3n)  (2m  -  3n).  23.  (4as  +  5y) 
(lax  -  by).  24.  (x  -  7)  (x  -  2).  25.  (x  -  7)  (s  +  1). 
26.   J  +  3. 


27.  as.  28.  a  -  J.  29.  a  +  3.  30.  a  +  1.  31.  3a  -  2. 
32.  63s7.  33.  12G0aV.  34.  4420aQ6V.  35.  18(s*  +  3s 
+  2).  36.  90(s*  -  1).  37.  3G(s'2  -  a2).  38.  a3  +  7a2  - 
a  -7. 


39.  ™ 

26'" 

40.  «?, 

4a 

s  +  2 
a  —  x,        — - . 

'    s  +  1 

41.  5-^. 

2U» 

.a    s  +  a 
42.  7  . 

a  —  o 

43.     ^,. 

a2  —  s2 

44.   t 

a  —  b 

IK             4;C 
45-     ^— 1  • 

46.      "     . 

s  —  2 

47.        «      . 

X"  —  1 

48.   -i%  . 

«2    —    1 

49.   3a  +  1. 
15 

50.   ^ —    -  and 

1  +  a 


51.  — 1— r  •       52.  4a  -  2.        53.  -, £- .       54.  1. 

a2  —  1  s-  —  9 

55.   -r-i =.     56.   6-^.       57.   4^.       58.  ^-. 

a    +  a  —  2  s  a  +  0  1  —  a 

59.  — .       60.  f  +  "  .       61.  ^|.      62.  5.       63.  2. 

a  2a  +  1  s  —  3 

64.  Ill,       65.  35.       66.   -  TV-       67.   -  19}.       68.  0. 
69    8.       70.  0. 


ANSWEBS.  155 


71.  5.         72. 

12.         73.  5}.         74. 

34f         75.    3. 

76.  1.        77.  16. 

78.  1.        79.  4. 

80.  12.      81.  7. 

82.   -2fi.        83. 

,  5.        84.  7ff.        85. 

144.        86.  3. 

87.  x  =  -  If) 

88.  a;  =    8) 

89.    x  =  5 ) 

y  =  -  n\ 

y  =  12) 

y=7)" 

90.  x  =  i) 

91.  x  =  12) 

92.  x  =  15 ) 

y  =  i)' 

2/  =  12)" 

y  =  14)' 

93.  x  =  4 ) 

94.  a;  =  19£) 

95.  a;  =    2^1 

y  =  6) 

y  =  -i> 

2T=    BJ-N 

s  =  10J 

96.  4.  97.  8.  98.  15,  16.  99.  65,  66. 

100.  180  dollars.  101.  ft,  20^.  102.  .0315,  .0385. 
103.  8TV  104.  30.  105.  $800,  83,200,  81,000. 

106.  20  inches,    16  inches.  107.  824,    872,    $24. 

108.   65^  miles  from  A,  13 4j  hours  after  starting. 

109.  81.  110.  80,  120.  111.  45  years  of  age,  and 
15  years  of  age.  112.  4,  7,  10,  13,  16  years  of  age  re- 
spectively. 113.  40,  20.  114.  7\,  14f,  8f  115.  -  2f 
116.  150.  117.  $120,000.  118.   &.  119.  12. 


156  ANSWERS. 

120.  425  feet,  450  feet.      121.  12  o'clock.     122.  15.     123. 

Eldest  son,  $12,467.52;  youngest  son,  19,350.64;  the 
widow,  $8,181.84.  124.  18  sheep,  16  oxen.  125.  9,  12. 
126.   6T\4T,  7fft.  127-  A,  $15,  and  B,  $18.  128. 

10,  30.  129.   20,  21.  130.   5T5r  minutes  past  one 

o'clock. 


131.  (1)  x*  -  6x*  -f  17z2  -  Ux  +  16  ;  (2)  16a*  -  16ab 
+  2iac  +  W  —  12bc  +  9c2  ;  (3)  x*  -  2ax  +  ibx  -  2cx  -f- 
a*  -  Aab  +  4fo2  +  2ac  -  Uc  +  c\  132.  7x  +  9a. 

133.  11«  -  156.       134.  20ax  -  5b.       135.  2a  -  36  +  5c. 
136.  x°-  -  2x  +  1. 


137.   ±  4.         138.   ±  I.         139.  ±  2.          140.   ±  2. 

141,  12,  or  -  6.          142.  9  or  -  1.  143.  15  or  -  14. 

144.  5  or  -  5|.             145.  b  ±  a.  146.  3  or  -  ^-. 

147.  15or-|.       148.  12,  16,  18.  149.  13  or  -14. 

150.  12,  9.         151.  30,  40.        152.  40,  45.          153.  14 
and  9,  or  18  and  7.         154.  24,  25. 


ANSWERS.  157 


155.  x  —  18  or  *&)  156.  x  =  9  or  -  14V 


y  =  3  or  —  | )  ^  =  4  or  —  6 


4-1    ) 

n 

4  ' 


157.  a  =  2  or  78        )  158.  x  =  6  or  114 


«! 


#  =  14  or  -  24 )  2/  =  1  or  -  1G1 

159.  5V27"  160.  4  +  2^/37  161.  13. 

162.  16ai>cV2a.  163.  |a/35\  164.  2(3  +  \/5). 

165.   ^/5  +  1.        166.  10  +  5V6.         167.  16.       168   5. 

169.  1. 


170.   15  :  16   greater   by  || ;    22  :  23   greater  by  -jfg. 

a  -i  ^o     25 

12c  '  56 


171.  =?-.  172.   ~.         173.  2.        174.  5    20. 


175.  7400.        176.  92i.         177.  Common  difference  f 
Series  12,  13-J,  etc.,  ...20.     178.  Common  difference   ^ 

Series  2J,  2T3T<V?  2fW,  etc->  •  •  •  2i-  1T9  135-  18°  14,;41- 
181.  a  +  x.  182.  -I,  f.  183.  36,  108,  324.  184.  ff 
185.   50625. 


18% 


